###  4.4 An example of a graded Hermitian norm on the section ring

The main goal of this section is emphasize that the equivalence class of a
graded Hermitian metric cannot be determined from the convergence of the
Fubini-Study operator.

######  Proposition 4.15.

There is a graded Hermitian norm $N=\sum_{k=1}^{\infty}N_{k}$ on
$R(\mathbb{P}^{1},\mathscr{O}(1))$, for which $FS(N_{k})^{\frac{1}{k}}$
converge uniformly, as $k\to\infty$, to a fixed smooth positive metric on
$\mathscr{O}(1)$, but $N$ is not equivalent to ${\rm{Hilb}}(h^{L})$ for any
continuous psh metric $h^{L}$ on $\mathscr{O}(1)$.

###### Proof.

Let us identify $\mathbb{P}^{1}$ to $\mathbb{P}(V^{*})$, where $V$ is a vector
space generated by two elements: $x$ and $y$. Let us consider a Hermitian
metric $H$ on $V$, which makes $x$ and $y$ an orthonormal basis, and denote by
$h^{FS}$ the induced Fubini-Study metric on $\mathscr{O}(1)$. Recall that for
any $k\in\mathbb{N}$, we have a classical isomorphism

${\rm{Sym}}^{k}(V)\to H^{0}(\mathbb{P}(V^{*}),\mathscr{O}(k)).$ (4.41)

Under this isomorphism, for any $a,b\in\mathbb{N}$, $a+b=k$, an easy
calculation shows that we have

$\big{\|}x^{a}\cdot
y^{b}\big{\|}_{{\rm{Hilb}}_{k}(h^{FS})}^{2}=\frac{a!b!}{(k+1)!}.$ (4.42)

We draw the attention of the reader that the above norm doesn’t coincide with
the norm $\|\cdot\|_{{\rm{Sym}}^{k}(H)}$ on ${\rm{Sym}}^{k}(V)$ induced by
$H$. In fact, an easy calculation gives

$\big{\|}x^{a}\cdot y^{b}\big{\|}_{{\rm{Sym}}^{k}(H)}^{2}=\frac{a!b!}{k!}.$
(4.43)

Let us consider a Hermitian norm $H_{k}$ on
$H^{0}(\mathbb{P}(V^{*}),\mathscr{O}(k))$, such that the basis $x^{a}\cdot
y^{b}$ is orthogonal and in the above notations, we have

$\big{\|}x^{a}\cdot y^{b}\big{\|}_{H_{k}}=\begin{cases}1,&\text{if }|a-b|\leq
1\\\ \big{\|}x^{a}\cdot
y^{b}\big{\|}_{{\rm{Hilb}}_{k}(h^{FS})},&\text{otherwise}.\end{cases}$ (4.44)

We will now verify that $H_{k}$ satisfies the assumptions of the proposition.
For simplicity, we only work with metrics $H_{k}$ with $k$ even. By Stirling’s
formula and (4.42), we have

$\frac{\big{\|}x^{k}\cdot y^{k}\big{\|}^{2}_{H_{2k}}}{\big{\|}x^{k}\cdot
y^{k}\big{\|}^{2}_{{\rm{Hilb}}_{2k}(h^{FS})}}\geq 2^{k}.$ (4.45)

Hence, the graded norm $\sum_{k=1}^{\infty}H_{k}$ is not equivalent to
${\rm{Hilb}}(h^{FS})$. We will now show that the Fubini-Study metric of
$H_{k}$ converges uniformly, as $k\to\infty$, to $h^{FS}$. This would clearly
imply by Theorem 2.15 that $\sum_{k=1}^{\infty}H_{k}$ satisfies the
assumptions of the proposition.

For this, from Lemma 2.14, we see that for any $a,b\in\mathbb{C}$, not
simultaneously equal to zero, the following identity between the Fubini-Study
metrics of $H_{k}$ and ${\rm{Hilb}}_{k}(h^{FS})$ holds

$1-\frac{FS({\rm{Hilb}}_{2k}(h^{FS}))}{FS(H_{2k})}\Big{(}[ax^{*}+by^{*}]\Big{)}=\Big{(}\frac{(2k)!}{k!k!}-1\Big{)}\cdot\frac{|ab|^{k}}{(|a|+|b|)^{2k}}.$
(4.46)

However, by mean inequality and Stirling’s formula, we obtain that there is
$C>0$, such that for any $k\in\mathbb{N}^{*}$, we have

$\Big{|}\Big{(}\frac{(2k)!}{k!k!}-1\Big{)}\cdot\frac{|ab|^{k}}{(|a|+|b|)^{2k}}\Big{|}\leq\frac{1}{\sqrt{k}}.$
(4.47)

From (4.46), (4.47) and Theorem 2.15, $FS(H_{k})^{\frac{1}{k}}$ converges
uniformly, as $k\to\infty$, to $h^{FS}$. ∎

As the following example of László Lempert shows, a phenomenon similar to
Proposition 4.15 holds in the non-asymptotic regime as well.

######  Proposition 4.16 (László Lempert).

There are sequences of Hermitian metrics $H_{\delta}$, $H^{\prime}_{\delta}$,
$\delta\in]0,1]$ over $H^{0}(\mathbb{P}^{1},\mathscr{O}(2))$, such that, as
$\delta\to\infty$, $\frac{FS(H_{\delta})}{FS(H^{\prime}_{\delta})}\to 1$, but
there is $c>0$ such that $d_{\infty}(H_{\delta},H^{\prime}_{\delta})>c$ for
any $\delta\in]0,1]$.

######  Remark 4.17.

See also Lempert [36] for a related result about injectivity of the Fubini-
Study map.

###### Proof.

We use similar notations as in the proof of Proposition 4.15. Let $H_{\delta}$
(resp. $H^{\prime}_{\delta}$) be so that the basis
$(s_{1,\delta},s_{2,\delta},s_{3,\delta})\mathrel{\mathop{\ordinarycolon}}=(xy,2xy+2\delta
y^{2},x^{2}+y^{2})$ (resp.
$(2s_{1,\delta},\frac{1}{2}s_{2,\delta},s_{3,\delta})$) is orthonormal with
respect to $H_{\delta}$ (resp. $H^{\prime}_{\delta}$). We now verify that such
choice satisfies the assumptions of the proposition. The second condition is
obvious since $H_{\delta}(s_{1,\delta},s_{1,\delta})=1$, but
$H^{\prime}_{\delta}(s_{1,\delta},s_{1,\delta})=\frac{1}{4}$. Now, from Lemma
2.14, we have

$\frac{FS(H_{\delta})}{FS(H^{\prime}_{\delta})}=\frac{4|s_{1,\delta}|^{2}+\frac{1}{4}|s_{2,\delta}|^{2}+|s_{3,\delta}|^{2}}{|s_{1,\delta}|^{2}+|s_{2,\delta}|^{2}+|s_{3,\delta}|^{2}},$
(4.48)

where $|\cdot|$ is any norm on $\mathscr{O}(2)$. From (4.48),
$\frac{FS(H_{\delta})}{FS(H^{\prime}_{\delta})}$ tends to $1$ uniformly, as
$\delta\to 0$. ∎

###  4.5 Inductive construction of the Hilbert map, a proof of Theorem 4.2

The main goal of this section is to establish Theorem 4.2. The proof is done
using the following result providing the inductive construction of the Hilbert
map.

Assume for simplicity that $L$ is very ample and the multiplication map
${\rm{Mult}}_{1,1}$ is surjective (hence, all multiplication maps are
surjective as well). Let $H_{1}$ be any Hermitian norm on $H^{0}(X,L)$. Then
using the map (1.2) by the surjectivity of the multiplication maps, one can
endow $H^{0}(X,L^{k})$ with the Hermitian norm
$H_{k}=[H_{1}\otimes\cdots\otimes H_{1}]$, where the tensor product is
repeated $k$ times. We introduce the graded Hermitian norm
$\mathcal{H}\mathrel{\mathop{\ordinarycolon}}=\sum_{k=1}^{\infty}H_{k}$ on
$R(X,L)$.

######  Theorem 4.18.

The graded Hermitian norms $\mathcal{H}$ and ${\rm{Hilb}}(FS(H_{1}))$ on
$R(X,L)$ are equivalent.

######  Remark 4.19.

a) The Fubini-Study operators for both of those norms coincide by Theorem 2.15
and Lemma 4.11. It doesn’t, however, imply the equivalence of the graded norms
by Section 4.4.

b) It is tempting to think that Theorem 4.18 follows from Lemma 4.9. This is,
however, not the case. Not only in Lemma 4.9, we require $k_{1},\cdots,k_{r}$
to be big enough, but also not every Hermitian norm on $H^{0}(X,L)$ lies in
the image of the Hilbert map, see Sun [53] for obstructions.

c) From Corollary B.7, $\mathcal{H}$ is multiplicatively generated. Hence,
Theorem 4.18 is a special case of Theorem 4.2 by Lemma 4.11. Our proof of
Theorem 4.2, however, passes by Theorem 4.18.

Before proceeding with the proof of Theorem 4.18, let us show how it implies
Theorem 4.2.

###### Proof of Theorem 4.2 assuming Theorem 4.18.

Let us fix $\epsilon>0$. By the uniform convergence of the sequence of metrics
$FS(N_{k})^{\frac{1}{k}}$, $k\in\mathbb{N}^{*}$, there is
$p_{1}\in\mathbb{N}$, such that for any $k\geq p_{1}$, we have

$d_{\infty}\big{(}FS(N_{k})^{\frac{1}{k}},FS(N)\big{)}<\frac{\epsilon}{5}.$
(4.49)

From the growth condition on $f$, we know that there is $p_{2}\in\mathbb{N}$,
such that for any $k\geq p_{2}$, we have

$|f(k)|\leq\frac{\epsilon}{5}k.$ (4.50)

Now, let us fix $p_{0}=\max\\{p_{1},p_{2}\\}$, $k\geq p_{0}$. We consider the
Hermitian metric $[N_{k}\otimes\cdots\otimes N_{k}]$ induced by the
multiplication map (1.2) on $H^{0}(X,L^{kr})$. The tensor product is repeated
here $r$ times. According to Theorem 4.18, there is $p_{3}\geq p_{0}$, such
that for any $r\geq p_{3}$, we have

$d_{\infty}\Big{(}[N_{k}\otimes\cdots\otimes
N_{k}],{\rm{Hilb}}_{rk}(FS(N_{k})^{\frac{1}{k}})\Big{)}<\frac{\epsilon}{5}r.$
(4.51)

From Corollary 2.11, (4.8), (4.49) and (4.51), there is $p_{4}\geq p_{3}$,
such that for any $r\geq p_{4}$, we have

$d_{\infty}\Big{(}[N_{k}\otimes\cdots\otimes
N_{k}],{\rm{Hilb}}_{rk}(FS(N))\Big{)}\leq\frac{2\epsilon}{5}rk.$ (4.52)

However, from (4.1), we conclude that

$N_{rk}\leq[N_{k}\otimes\cdots\otimes
N_{k}]\cdot\exp\Big{(}\frac{\epsilon}{5}rk\Big{)}.$ (4.53)

In total, from (4.52) and (4.53), we deduce that

$N_{rk}\leq{\rm{Hilb}}_{rk}(FS(N))\cdot\exp\Big{(}\frac{3\epsilon}{5}rk\Big{)}.$
(4.54)

Now, since $k\in\mathbb{N}$ is fixed, and the spaces $H^{0}(X,L^{p})$,
$p=k,\ldots,2k-1$, are finitely dimensional, the norms $N_{p}$ and
${\rm{Hilb}}_{p}(FS(N))$ are comparable up to a uniform constant. From this
and Theorem 1.5, there is $p_{5}\geq p_{4}$, such that for any $0\leq l\leq
k-1$, $r\geq p_{5}$, we have

$d_{\infty}\Big{(}[{\rm{Hilb}}_{rk}(FS(N))\otimes
N_{k+l}],{\rm{Hilb}}_{(r+1)k+l}(FS(N))\Big{)}\leq\frac{\epsilon}{5}((r+1)k+l).$
(4.55)

On another hand, from Definition 1.3 and (4.50), we conclude that

$N_{(r+1)k+l}\leq[N_{rk}\otimes
N_{k+l}]\cdot\exp\Big{(}\frac{\epsilon}{5}((r+1)k+l)\Big{)}.$ (4.56)

Now, from (4.54), (4.55) and (4.56), we deduce that for any $k\geq
2p_{5}p_{0}$, we have

$N_{k}\leq{\rm{Hilb}}_{k}(FS(N))\cdot\exp(\epsilon k).$ (4.57)

Directly from Lemma 2.14, we see that for any $k\in\mathbb{N}^{*}$, we have

$N_{k}\geq{\rm{Ban}}^{\infty}_{k}(FS(N_{k})^{\frac{1}{k}}).$ (4.58)

From Corollary 2.11, (4.49) and (4.58), there is $p_{6}\in\mathbb{N}^{*}$,
such that for any $k\geq p_{6}$, we have

$N_{k}\geq{\rm{Hilb}}_{k}(FS(N))\cdot\exp(-\epsilon k).$ (4.59)

As $\epsilon>0$ was arbitrary, the statement follows from (4.57) and (4.59). ∎

###### Proof of Theorem 4.18.

Our proof is essentially based on two ingredients. First, we need to verify
Theorem 4.18 for the projective space by an explicit calculation. Second, we
apply the semiclassical version of Ohsawa-Takegoshi extension theorem, Theorem
2.6, for the Kodaira embedding to extend Theorem 4.18 from projective spaces
to general manifolds.

Let us first do the calculations for the projective space
$X\mathrel{\mathop{\ordinarycolon}}=\mathbb{P}(V^{*})$, where $V$ is a fixed
vector space of dimension $n$. In this case, we take
$L\mathrel{\mathop{\ordinarycolon}}=\mathscr{O}(1)$. Then through the
identification of $H^{0}(X,L)$ with $V$, we may view the Hermitian metric
$H_{1}$ on $H^{0}(X,L)$ as a Hermitian metric on $V$. We induce the Fubini-
Study metric $FS(H_{1})$ on $L$ from this identification.

A direct calculation shows that the Hermitian norm
$H_{k}\mathrel{\mathop{\ordinarycolon}}=[H_{1}\otimes\cdots\otimes H_{1}]$
corresponds to the norm ${\rm{Sym}}^{k}(H_{1})$ induced on ${\rm{Sym}}^{k}(V)$
by $H_{1}$ under the identification (4.41). Now, as it was already exemplified
during the proof of Proposition 4.15, see (4.42), this norm is not isometric
to ${\rm{Hilb}}_{k}(FS(H_{1}))$, but the following identity holds

${\rm{Hilb}}_{k}(FS(H_{1}))=\sqrt{\frac{k!}{(k+n-1)!}}\cdot{\rm{Sym}}^{k}(H_{1}).$
(4.60)

Note that the factor in (4.60) grows polynomially in $k$ (in particular, it is
subexponential). Hence, Theorem 4.18 holds in this particular case.

Now, to establish Theorem 4.18 in its full generality, let us consider the
Kodaira embedding ${\rm{Kod}}_{1}$ from (2.28). We denote by
${\rm{Res}}_{{\rm{Kod}},X}$ the associated restriction operator

${\rm{Res}}_{{\rm{Kod}},k}\mathrel{\mathop{\ordinarycolon}}H^{0}(\mathbb{P}(H^{0}(X,L)^{*}),\mathscr{O}(k))\to
H^{0}(X,L^{k}).$ (4.61)

The crux of the matter is to realize that the multiplication operator
${\rm{Mult}}_{1,\cdots,1}$ from (1.2) factorizes under the identification
(4.41) of $H^{0}(\mathbb{P}(H^{0}(X,L)^{*}),\mathscr{O}(k))$ with
${\rm{Sym}}^{k}(H^{0}(X,L))$ through the symmetrization, ${\rm{Sym}}$, and the
restriction operator as follows

${H^{0}(X,L)^{\otimes
k}}$${{\rm{Sym}}^{k}(H^{0}(X,L))}$${H^{0}(\mathbb{P}(H^{0}(X,L)^{*}),\mathscr{O}(k))}$${H^{0}(X,L^{k}).}$$\scriptstyle{{\rm{Mult}}_{1,\cdots,1}}$$\scriptstyle{\rm{Sym}}$$\scriptstyle{{\rm{Res}}_{{\rm{Kod}},k}}$
(4.62)

Now, the proof goes as follows. From the first part of the proof of Theorem
4.18, we obtain that the quotient metric on ${\rm{Sym}}^{k}(H^{0}(X,L))$
associated to $H\otimes\cdots\otimes H$ is equivalent under the isomorphism
(4.41) to the metric
${\rm{Hilb}}_{k}^{\mathbb{P}(H^{0}(X,L)^{*})}(FS(H_{1}))$. Now, by the
semiclassical version of Ohsawa-Takegoshi extension theorem, see Theorem 2.6
or Corollary 2.12, the quotient of the metric
${\rm{Hilb}}_{k}^{\mathbb{P}(H^{0}(X,L)^{*})}(FS(H_{1}))$ under the map
${\rm{Res}}_{{\rm{Kod}},k}$ is equivalent to ${\rm{Hilb}}_{k}^{X}(FS(H_{1}))$.
But from the commutative diagram (4.62), the quotient metric
$[H\otimes\cdots\otimes H]$ on $H^{0}(X,L^{k})$ can be obtained by two
subsequent quotients: one through the map ${\rm{Sym}}$, another through the
map ${\rm{Res}}_{{\rm{Kod}},k}$, and the above reasoning shows that the
subsequent quotient is equivalent to ${\rm{Hilb}}_{k}^{X}(FS(H_{1}))$. ∎

###  4.6 Mabuchi geodesics and geodesics on the space of norms

The main goal of this section is to prove a relation between Mabuchi geodesics
and geodesics in the space of norms, announced in Theorem 1.8, from which we
borrow the notation. Let us first recall the following result.

######  Theorem 4.20.

For any $t\in[0,1]$, $FS(H_{k,t})^{\frac{1}{k}}$ converges uniformly to
$h^{L}_{t}$, as $k\to\infty$. Moreover, this convergence is uniform in $t$.

###### Proof.

For smooth extremities $h^{L}_{0},h^{L}_{1}$, this result was established by
Phong-Sturm [44, Theorem 1] for a slightly weaker than uniform convergence and
by Berndtsson [6, Theorem 1.2] for the uniform convergence. We argue that the
proof of Berndtsson works more generally for
$h^{L}_{0},h^{L}_{1}\in\mathcal{H}^{L}$. In fact, the regularity is used only
in the first part of the proof of [6, Theorem 4.2], refining [6, Theorem 1.2],
where author relies on the diagonal asymptotic expansion of the Bergman kernel
to estimate from above the logarithm of the Bergman kernel of $H_{k,t}$. This
argument can be replaced by Theorem 2.15 at the cost of replacing the
$O(\frac{\log k}{k})$-estimate by an $o(1)$-estimate, which does not affect
the final conclusion. ∎

Now, to establish Theorem 1.8, we will need several basic fact from linear
algebra. Let $U$ and $V$ be two finitely dimensional vector spaces and let
$f\mathrel{\mathop{\ordinarycolon}}U\to V$ be any map between them. Recall
that we say that $f$ is contracting with respect to the norms $\|\cdot\|_{U}$
on $U$ and $\|\cdot\|_{V}$ on $V$ (or with respect to the pair
$\|\cdot\|_{U}$, $\|\cdot\|_{V}$) if for any $x\in U$, we have
$\|\pi(x)\|_{V}\leq\|x\|_{U}$, or, in other words $\|\pi\|\leq 1$, where
$\|\cdot\|$ is the operator norm. Let $H^{U}_{0}$, $H^{U}_{1}$ (resp.
$H^{V}_{0}$, $H^{V}_{1}$) be two Hermitian metrics on $U$ (resp. $V$). Let us
denote by $H^{U}_{t}$, $H^{V}_{t}$, $t\in[0,1]$, the geodesics between
$H^{U}_{0}$, $H^{U}_{1}$ and $H^{V}_{0}$, $H^{V}_{1}$ in the space of
Hermitian norms on $U$ and $V$ respectively.

######  Lemma 4.21.

Assume that the map $\pi$ is contracting with respect to both pairs
$H^{U}_{0},H^{V}_{0}$ and $H^{U}_{1},H^{V}_{1}$. Then it is contracting with
respect to the pair $H^{U}_{t},H^{V}_{t}$ for any $t\in[0,1]$.

######  Remark 4.22.

By choosing a simultaneous diagonalization of $H^{U}_{0}$, $H^{U}_{1}$, we may
identify $(V,H^{V}_{0})$ (resp. $(V,H^{V}_{1})$) to the space
$L^{2}(\\{1,2,\cdots,\dim V\\},\mu_{V})$ (resp. $L^{2}(\\{1,2,\cdots,\dim
V\\},\omega_{V}\mu_{V})$), where $\mu_{V}=\delta_{1}+\cdots+\delta_{\dim V}$
is the counting measure and
$\omega_{V}\mathrel{\mathop{\ordinarycolon}}\\{1,2,\cdots,\dim
V\\}\to]0,+\infty[$ is a certain weight function. Similarly, we may identify
$(U,H^{U}_{0})$ (resp. $(U,H^{U}_{1})$) to the space $L^{2}(\\{1,2,\cdots,\dim
U\\},\mu_{U})$ (resp. $L^{2}(\\{1,2,\cdots,\dim U\\},\omega_{U}\mu_{U})$),
where similarly $\mu_{U}=\delta_{1}+\cdots+\delta_{\dim U}$,
$\omega_{U}\mathrel{\mathop{\ordinarycolon}}\\{1,2,\cdots,\dim
V\\}\to]0,+\infty[$. The result now follows from the interpolation theorem of
Stein-Weiss, see [4, Theorem 5.4.1].

######  Corollary 4.23.

Let us fix a surjective map $\pi\mathrel{\mathop{\ordinarycolon}}V\to Q$
between vector spaces $V,Q$. Let $N^{V}_{0}$, $N^{V}_{1}$ be two Hermitian
norms on $V$ and $N^{Q}_{0}$, $N^{Q}_{1}$ be the induced quotient Hermitian
norms on $Q$. For $t\in[0,1]$, we denote by $N^{V}_{t}$ the geodesics between
$N^{V}_{0}$ and $N^{V}_{1}$, and by $N^{U}_{t}$ the geodesics between
$N^{U}_{0}$ and $N^{U}_{1}$. Then for any $t\in[0,1]$, we have

$[N^{V}_{t}]\geq N^{U}_{t}.$ (4.63)

Let us now denote by $N^{V,0}_{0}$, $N^{V,0}_{1}$ two other Hermitian norms on
$V$, verifying $N^{V}_{0}\geq N^{V,0}_{0}$ and $N^{V}_{1}\geq N^{V,0}_{1}$.
Similarly, we denote by $N^{V,0}_{t}$, $t\in[0,1]$ the geodesics between
$N^{V,0}_{0}$ and $N^{V,0}_{1}$. Then for any $t\in[0,1]$, we have

$N^{V}_{t}\geq N^{V,0}_{t}.$ (4.64)

###### Proof.

It follows directly from Lemma 4.21. ∎

###### Proof of Theorem 1.8..

Let us denote by $f_{0}$ and $f_{1}$ the functions from Definition 1.3
associated to ${\rm{Hilb}}(h^{L}_{0})$, ${\rm{Hilb}}(h^{L}_{1})$ respectively
by Theorem 1.5. We will now establish that for any $t\in[0,1]$, the inequality
(4.1) holds for $N_{k}\mathrel{\mathop{\ordinarycolon}}=H_{k,t}$ and
$f\mathrel{\mathop{\ordinarycolon}}=f_{t}\mathrel{\mathop{\ordinarycolon}}=(1-t)f_{0}+tf_{1}$.
Remark first that $H_{k_{1},t}\otimes\cdots\otimes H_{k_{r},t}$, $t\in[0,1]$,
is a geodesic between
${\rm{Hilb}}_{k_{1}}(h^{L}_{0})\otimes\cdots\otimes{\rm{Hilb}}_{k_{r}}(h^{L}_{0})$
and
${\rm{Hilb}}_{k_{1}}(h^{L}_{1})\otimes\cdots\otimes{\rm{Hilb}}_{k_{r}}(h^{L}_{1})$
in the space of Hermitian norms on $H^{0}(X,L^{k_{1}})\otimes\cdots\otimes
H^{0}(X,L^{k_{r}})$.

From this and the first part of Corollary 4.23, we deduce that if we denote by
$H_{k_{1},\cdots,k_{r};t}$, $t\in[0,1]$, the geodesic between
$[{\rm{Hilb}}_{k_{1}}(h^{L}_{0})\otimes\cdots\otimes{\rm{Hilb}}_{k_{r}}(h^{L}_{0})]$
and
$[{\rm{Hilb}}_{k_{1}}(h^{L}_{1})\otimes\cdots\otimes{\rm{Hilb}}_{k_{r}}(h^{L}_{1})]$
in the space of Hermitian norms on $H^{0}(X,L^{k})$, then for any $t\in[0,1]$,
we have

$H_{k_{1},\cdots,k_{r};t}\leq[H_{k_{1},t}\otimes\cdots\otimes H_{k_{r},t}].$
(4.65)

Now, from the definitions of $f_{0}$, $f_{1}$ and the second part of Corollary
4.23, we deduce that

$H_{k,t}\leq
H_{k_{1},\cdots,k_{r};t}\cdot\exp(f_{t}(k_{1})+\cdots+f_{t}(k_{r})+f_{t}(k)).$
(4.66)

From (4.65) and (4.66), we deduce that the inequality (4.1) holds for
$N_{k}\mathrel{\mathop{\ordinarycolon}}=H_{k,t}$ and
$f\mathrel{\mathop{\ordinarycolon}}=f_{t}\mathrel{\mathop{\ordinarycolon}}=(1-t)f_{0}+tf_{1}$.
From this, Theorem 4.2 and Theorem 4.20, we conclude that for any $t\in[0,1]$,
the graded norm $H_{t}$ is equivalent to ${\rm{Hilb}}(h^{L}_{t})$. ∎

######  Remark 4.24.

A quick inspection of the proof of Theorem 4.2 shows that the equivalence of
$H_{t}$ to ${\rm{Hilb}}(h^{L}_{t})$ is uniform in $t\in[0,1]$ (i.e. the
convergence $\frac{1}{k}d_{\infty}(H_{k,t},{\rm{Hilb}}_{k}(h^{L}_{t}))\to 0$
from (1.13) is uniform in $t$). This is due to the fact that the function
$f_{t}$ above is uniformly bounded by a function
$f_{\max}\mathrel{\mathop{\ordinarycolon}}=\max\\{f_{0},f_{1}\\}$, satisfying
the growth assumption $f_{\max}(k)=o(k)$, and the convergence from Theorem
4.20 is uniform. Hence, Theorem 1.8 refines Theorem 4.20 by Theorem 2.15,
Lemma 4.12 and Proposition 4.15.

##  Appendix A On almost sub-additive sequences of functions

The main goal of this section is to study almost sub-additive sequences. More
precisely, in Section A.1, we state the analogue of Fekete’s lemma. In Section
A.2, we study the uniform convergence of sub-additive sequences of functions
and state the analogue of Dini’s theorem.

###  A.1 A version of Fekete’s lemma for almost sub-additive sequences

In this section we discuss a generalization of Fekete’s lemma, cf. [26],
studying sub-additive sequences of real numbers. We will show that the
conclusion of the lemma is still valid if the sequence is sub-additive up to a
small error term.

Recall that Bruijn-Erdős in [19, Theorem 23] considered sequences for which
the sub-additivity condition $a_{m+n}\leq a_{m}+a_{n}$ is replaced by a weaker
one

$a_{m+n}\leq a_{m}+a_{n}+g_{m+n}.$ (A.1)

As they show, a conclusion of Fekete’s lemma only holds under the assumption

$\sum\frac{g_{n}}{n^{2}}<+\infty.$ (A.2)

As this condition doesn’t appear quite naturally in geometric setting, such as
Bernstein-Markov property, we find it unsuitable for the encapsulation of the
features of the $L^{2}$-norm as we do it in Definition 1.3. We, thus, consider
an alternative condition, which demands more from the sequence $a_{n}$ and
less from the error term.

######  Definition A.1.

We say that a sequence of numbers $a_{n}\in\real$, $n\in\mathbb{N}$, is almost
sub-additive if for any $r\geq 2$, $k;k_{1},\ldots,k_{r}\in\mathbb{N}$,
$k_{1},\ldots,k_{r}\geq p_{0}$, $k_{1}+\cdots+k_{r}=k$, the following
inequality holds

$a_{k}\leq a_{k_{1}}+\cdots+a_{k_{r}}+f_{k_{1}}+\cdots+f_{k_{r}}+f_{k},$ (A.3)

where $f_{k}$, verifies $f_{k}=o(k)$, as $k\to\infty$. We say that $a_{k}$ is
almost super-additive if the above inequality holds with a reverse sign. In
the latter case we denote the sequence $f_{k}$ by $g_{k}$, $k\in\mathbb{N}$.

######  Lemma A.2 (Chen [12, Proposition 3.1]).

Assume that a sequence $a_{n}$, $n\in\mathbb{N}$, is almost sub-additive and
non-negative. Then the sequence $\frac{a_{n}}{n}$ has a finite limit, as
$n\to\infty$.

######  Corollary A.3.

Assume that a sequence $a_{n}$, $n\in\mathbb{N}$, is both almost sub-additive
and almost super-additive. Then the sequence $\frac{a_{n}}{n}$ has a finite
limit, as $n\to\infty$.

###### Proof.

Since $a_{n}$ is super-additive, we have

$a_{n}\geq nc_{n},$ (A.4)

where $c_{n}$ is defined as $c_{n}=a_{1}+\frac{g_{n}}{n}$ for a sequence
$g\mathrel{\mathop{\ordinarycolon}}\mathbb{N}\to\real$ from Definition A.1.
Clearly, by our assumption on $g_{n}$, the sequence $c_{n}$ is bounded from
below by $c\in\real$. We can now apply Lemma A.2 for a sequence $a_{n}-nc$ to
establish the result. ∎

###  A.2 Uniform convergence of almost sub-additive sequences of functions

The main goal of this section is to study the uniform convergence of almost
sub-additive sequences of functions. The following definition will be
particularly useful later on.

######  Definition A.4.

A sequence of functions $a_{k}\mathrel{\mathop{\ordinarycolon}}Y\to\real$,
$k\in\mathbb{N}^{*}$, defined over a topological space $Y$ will be called
uniformly almost sub-additive if for any $x\in Y$, the sequence $a_{k}(x)$ is
almost sub-additive, and in the definition of almost sub-additivity the
sequences $f_{n}\mathrel{\mathop{\ordinarycolon}}=f_{n}(x)$ can be chosen in
such way that the convergence $\frac{f_{k}}{k}\to 0$ holds locally uniformly
on $Y$, as $k\to\infty$. Similarly, we define the notion of uniformly almost
super-additive sequences of functions.

The following lemma is an analogue of Dini’s theorem in our setting.

######  Lemma A.5.

Assume that the sequence of non-negative functions
$a_{k}\mathrel{\mathop{\ordinarycolon}}Y\to\real$, $k\in\mathbb{N}^{*}$,
defined over a compact space $Y$, is uniformly almost sub-additive. Assume
also that the pointwise limit $a\mathrel{\mathop{\ordinarycolon}}Y\to\real$ of
$\frac{a_{k}}{k}$, as $k\to\infty$, which exists by Lemma A.2, is a continuous
function. Then the convergence of $\frac{a_{k}}{k}$ to $a$ is uniform.

###### Proof.

A straightforward modification of the proof of Dini’s theorem. ∎

The next lemma is an almost-subadditive analogue of a well-known statement for
decreasing sequences of functions.

######  Lemma A.6.

Let $Y$ be any topological space. Assume
$a_{k}\mathrel{\mathop{\ordinarycolon}}Y\to\real$, $k\in\mathbb{N}^{*}$, is a
sequence of positive continuous functions which is uniformly almost sub-
additive. Then the pointwise limit
$a\mathrel{\mathop{\ordinarycolon}}Y\to\real$ of $\frac{a_{k}}{k}$, as
$k\to\infty$, which exists by Lemma A.2, is upper semicontinuous.

###### Proof.

A straightforward modification of the proof for decreasing sequences of
functions. ∎

######  Lemma A.7.

Let $Y$ be a compact topological space. Assume that
$a_{k}\mathrel{\mathop{\ordinarycolon}}Y\to\real$, $k\in\mathbb{N}^{*}$, is a
sequence of continuous functions which is uniformly almost sub-additive and
uniformly almost super-additive at the same time. Then the sequence
$\frac{a_{k}}{k}$ has a continuous limit $a$, and the convergence of
$\frac{a_{k}}{k}$ to $a$ is uniform.

###### Proof.

Uniform almost super-additivity implies that there is $C>0$ such that
$a_{k}\geq-Ck$, see (A.4) and the paragraph after for the details. An easy
verification shows that the sequence
$h_{k}\mathrel{\mathop{\ordinarycolon}}=a_{k}+Ck$ then satisfies all the
assumptions of Lemma A.6 and hence the limit $a$ is upper semicontinuous. We
repeat the same argument for the sequence $-a_{k}$ to obtain that $a$ is lower
semicontinuous. Hence, $a$ is continuous. The result now follows from Lemma
A.5, applied to $h_{k}$. ∎

##  Appendix B Normed vector spaces and induced tensor norms

The main goal of this section is to recall some classical results about normed
vector spaces and the induced norms on the tensor products.

The first result is about the identity of two natural norms, defined on the
quotient of a normed vector space. Let $(U,N_{U}=\|\cdot\|_{U})$ be a normed
vector space. For any quotient $\pi\mathrel{\mathop{\ordinarycolon}}U\to Q$,
consider the following two metrics. First, we have a metric
$\|\cdot\|_{Q}^{0}$ induced by $\|\cdot\|_{U}$ and the quotient map as in
(1.9). Second, we have a metric $\|\cdot\|_{Q}^{1}$, given by the dual of the
norm on $Q^{*}$ induced by the dual metric $\|\cdot\|_{U}^{*}$ and the
inclusion $\pi^{*}\mathrel{\mathop{\ordinarycolon}}Q^{*}\to U^{*}$.

######  Lemma B.1.

The following identity holds $\|\cdot\|_{Q}^{0}=\|\cdot\|_{Q}^{1}$.

###### Proof.

A trivial verification. ∎

The next result is about decomposable tensors and their injective and
projective norms. Let $(U,N_{U}=\|\cdot\|_{U})$, $(V,N_{V}=\|\cdot\|_{V})$ be
two Banach spaces. Recall that the projective tensor product norm
$N_{U}\otimes_{\pi}N_{V}=\|\cdot\|_{\otimes_{\pi}}$, and the injective tensor
product norm $N_{U}\otimes_{\epsilon}N_{V}=\|\cdot\|_{\otimes_{\epsilon}}$
were defined in (1.7) and (1.8) respectively. We will also sometimes consider
the completion $(U\hat{\otimes}_{\epsilon}V,N_{U}\otimes_{\epsilon}N_{V})$
(resp. $(U\hat{\otimes}_{\pi}V,N_{U}\otimes_{\pi}N_{V})$) of $(U\otimes
V,N_{U}\otimes_{\epsilon}N_{V})$ (resp. $(U\otimes
V,N_{U}\otimes_{\pi}N_{V})$).

######  Lemma B.2.

For any $x\in U$, $y\in V$, we have $\|x\otimes y\|_{\otimes_{\pi}}=\|x\otimes
y\|_{\otimes_{\epsilon}}=\|x\|_{U}\cdot\|y\|_{V}$. Also for any $\phi\in
U^{*}$, $\psi\in V^{*}$, we have similar identities for the induced dual norms
$\|\phi\otimes\psi\|_{\otimes_{\pi}}^{*}=\|\phi\otimes\psi\|_{\otimes_{\epsilon}}^{*}=\|\phi\|_{U}^{*}\cdot\|\psi\|_{V}^{*}$.
Moreover, any norm on $U\otimes V$, which satisfies those properties lies in
between $N_{U}\otimes_{\epsilon}N_{V}$ and $N_{U}\otimes_{\pi}N_{V}$.

###### Proof.

For the projective (resp. injective) norm, the first part of the statement is
proved in [49, Propositions 2.1 and 2.3] (resp. [49, Proposition 3.1]). The
second part of the statement is established in [49, Proposition 6.1]. ∎

Now let us recall the following standard duality property in finitely
dimensional context, which helps to pass from injective tensor product to the
projective one.

######  Lemma B.3 ([49, Theorem 4.21]).

Assume that both $U$ and $V$ are finitely dimensional. The following identity
between the norms on $U^{*}\otimes V^{*}$ holds

$N_{U}^{*}\otimes_{\pi}N_{V}^{*}=(N_{U}\otimes_{\epsilon}N_{V})^{*}.$ (B.1)

If, moreover, both $N_{U}$ and $N_{V}$ are Hermitian norms, then

$N_{U}^{*}\otimes N_{V}^{*}=(N_{U}\otimes N_{V})^{*}.$ (B.2)

######  Remark B.4.

In [49, Theorem 4.21], author even gives a necessary condition so that the
above identity holds in the infinite dimensional context, if one takes the
Banach closure of each side of (B.1).

The following lemma gives a justification for the names injective and
projective tensor norms. Let $Q$ be a quotient of $U$. Endow it with the
quotient norm $N_{Q}=\|\cdot\|_{Q}=[N_{U}]$ as defined in (1.9). Let $W$ be a
closed subspace of $U$. Endow it with the induced norm $N_{W}=\|\cdot\|_{W}$.

######  Lemma B.5.

The norm on $Q\otimes V$ induced by $N_{U}\otimes_{\pi}N_{V}$ and the quotient
map $U\hat{\otimes}_{\pi}V\to Q\otimes V$ (or $U\otimes_{\pi}V\to Q\otimes V$)
coincides with the norm $N_{Q}\otimes_{\pi}N_{V}$. Similarly, the norm on
$W\otimes V$ induced by $N_{U}\otimes_{\epsilon}N_{V}$ and the induced map
$W\otimes V\to U\hat{\otimes}_{\epsilon}V$ (or $W\otimes V\to
U\otimes_{\epsilon}V$) coincides with the norm $N_{W}\otimes_{\epsilon}N_{V}$.
When both $N_{U}$ and $N_{V}$ come from Hermitian tensor products, the
analogous statements hold for the Hermitian tensor product.

###### Proof.

The statement about the projective (resp. injective) tensor norm is
established in [49, Proposition 2.5] (resp. [49, Proposition 3.2 and the
discussion after]). For Hermitian norms, the proof is direct. ∎

Let $(V_{i},N_{i}=\|\cdot\|_{i})$, $i=1,2,3$, be finitely dimensional normed
vector spaces. Our next result concerns the associativity property for
injective and projective tensor norms.

######  Lemma B.6.

On the vector space $V_{1}\otimes V_{2}\otimes V_{3}$ we have the following
identity between the norms

$(N_{1}\otimes_{\epsilon}N_{2})\otimes_{\epsilon}N_{3}=N_{1}\otimes_{\epsilon}(N_{2}\otimes_{\epsilon}N_{3}),\quad(N_{1}\otimes_{\pi}N_{2})\otimes_{\pi}N_{3}=N_{1}\otimes_{\pi}(N_{2}\otimes_{\pi}N_{3}).$
(B.3)

In particular, the notations
$N_{1}\otimes_{\epsilon}N_{2}\otimes_{\epsilon}N_{3}$,
$N_{1}\otimes_{\pi}N_{2}\otimes_{\pi}N_{3}$ are well-defined.

###### Proof.

Follows directly from the definitions. ∎

Let us establish the following transitivity property for the quotients of
projective tensor norm. Let $Q_{3}$ be a quotient of $V_{1}\otimes V_{2}$ and
$Q$ be a quotient of $V_{1}\otimes V_{2}\otimes V_{3}$, which factorizes
through the natural map $V_{1}\otimes V_{2}\otimes V_{3}\to Q_{3}\otimes
V_{3}$.

######  Corollary B.7.

On the vector space $Q$, we have the following identity between the norms

$\big{[}[N_{1}\otimes_{\pi}N_{2}]\otimes_{\pi}N_{3}\big{]}=\big{[}N_{1}\otimes_{\pi}N_{2}\otimes_{\pi}N_{3}\big{]},$
(B.4)

where we used the notation for the quotient norm introduced in (1.9).
Moreover, if $N_{1}$, $N_{2}$, $N_{3}$ come from Hermitian products, then the
analogous property holds for the Hermitian tensor product

$\big{[}[N_{1}\otimes N_{2}]\otimes N_{3}\big{]}=\big{[}N_{1}\otimes
N_{2}\otimes N_{3}\big{]}.$ (B.5)

###### Proof.

It follows trivially from Lemmas B.5 and B.6. ∎

On several occasions we need to compare injective and projective tensor norms.
Let $(V_{i},N_{i}=\|\cdot\|_{i})$, $i=1,\ldots,N$, $N\in\mathbb{N}^{*}$ be
finitely dimensional normed vector spaces.

######  Lemma B.8.

The following inequality between the norms on $V_{1}\otimes\cdots\otimes
V_{N}$ holds

$N_{1}\otimes_{\pi}\cdots\otimes_{\pi}N_{N}\leq
N_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}N_{N}\cdot\frac{\dim
N_{1}\cdot\ldots\cdot\dim N_{N}}{\max\\{\dim N_{1},\cdots,\dim N_{N}\\}}.$
(B.6)

###### Proof.

It is classical, cf. [34, Theorem 21], that (B.6) holds for $N=2$. The general
result now follows from this special case by induction and Lemma B.6. ∎

We will finally calculate explicitly some examples of injective and projective
tensor norms. In what follows, we consider vector space n, endowed with the
$L^{1}$, $L^{2}$ and $L^{\infty}$-norms, which we denote below in a slightly
non-standard manner as $l^{1}_{n}$, $l^{2}_{n}$, $l^{\infty}_{n}$. We fix also
two compact topological spaces $K_{1},K_{2}$ and denote by
$(\mathscr{C}^{0}(K_{i}),\|\cdot\|_{L^{\infty}_{i}})$, $i=1,2$, the associated
spaces of continuous functions on $K_{i}$, endowed with the supremum norm. We
consider the analogous functional space on the product space
$(\mathscr{C}^{0}(K_{1}\times K_{2}),\|\cdot\|_{L^{\infty}_{1\times 2}})$. Let
us also fix two measured spaces $(X_{i},\mu_{i})$, $i=1,2$, and consider the
associated spaces of $L^{1}$ and $L^{2}$-functions
$(L^{q}(X_{i}),\|\cdot\|_{L^{q}(\mu_{i})})$, $q=1,2$. We consider the product
measure $\mu_{1}\times\mu_{2}$ on $X_{1}\times X_{2}$ and the associated
spaces of $L^{1}$ and $L^{2}$-functions, $(L^{q}(X_{1}\times
X_{2}),\|\cdot\|_{L^{q}(\mu_{1}\times\mu_{2})})$, $q=1,2$.

######  Lemma B.9.

The following identities between Banach spaces hold

$\displaystyle\big{(}L^{1}(X_{1}),\|\cdot\|_{L^{1}(\mu_{1})}\big{)}\hat{\otimes}_{\pi}\big{(}L^{1}(X_{2}),\|\cdot\|_{L^{1}(\mu_{2})}\big{)}$
$\displaystyle=\big{(}L^{1}(X_{1}\times
X_{2}),\|\cdot\|_{L^{1}(\mu_{1}\times\mu_{2})}\big{)},$ (B.7)
$\displaystyle\big{(}L^{2}(X_{1}),\|\cdot\|_{L^{2}(\mu_{1})}\big{)}\otimes\big{(}L^{2}(X_{2}),\|\cdot\|_{L^{2}(\mu_{2})}\big{)}$
$\displaystyle=\big{(}L^{2}(X_{1}\times
X_{2}),\|\cdot\|_{L^{2}(\mu_{1}\times\mu_{2})}\big{)},$
$\displaystyle\big{(}\mathscr{C}^{0}(K_{1}),\|\cdot\|_{L^{\infty}_{1}}\big{)}\hat{\otimes}_{\epsilon}\big{(}\mathscr{C}^{0}(K_{2}),\|\cdot\|_{L^{\infty}_{2}}\big{)}$
$\displaystyle=\big{(}\mathscr{C}^{0}(K_{1}\times
K_{2}),\|\cdot\|_{L^{\infty}_{1\times 2}}\big{)},$

where the tensor product in the second line means the tensor product of
Hilbert spaces. In particular, under the natural isomorphism
${}^{n}\otimes{}^{m}\to{}^{n\cdot m}$, for norms on ${}^{n}\otimes{}^{m}$, we
have

$l^{1}_{n}\otimes_{\pi}l^{1}_{m}=l^{1}_{n\cdot m},\qquad l^{2}_{n}\otimes
l^{2}_{m}=l^{2}_{n\cdot m},\qquad
l^{\infty}_{n}\otimes_{\epsilon}l^{\infty}_{m}=l^{\infty}_{n\cdot m}.$ (B.8)

###### Proof.

Remark first that the identities (B.8) follow directly from (B.7) by applying
the result for discrete spaces. Now, the first (resp. last) identity from
(B.7) is from Ryan [49, Exercise 2.8] (resp. Ryan [49, §3.2, p.50]). The
middle identity from (B.7) is standard. ∎

## References

  * [1] G. Aubrun, L. Lami, C. Palazuelos, S. J. Szarek, and A. Winter.  Universal gaps for XOR games from estimates on tensor norm ratios.  Commun. Math. Phys., 375(1):679–724, 2020.
  * [2] E. Bedford and B. A. Taylor.  The Dirichlet problem for a complex Monge-Ampère equation.  Invent. Math., 37:1–44, 1976.
  * [3] E. Bedford and B. A. Taylor.  A new capacity for plurisubharmonic functions.  Acta Math., 149:1–40, 1982.
  * [4] J. Bergh and J. Löfström.  Interpolation spaces. An introduction, volume 223 of Grundlehren Math. Wiss. 1976\.
  * [5] R. Berman, S. Boucksom, and D. Witt Nyström.  Fekete points and convergence towards equilibrium measures on complex manifolds.  Acta Math., 207(1):1–27, 2011.
  * [6] B. Berndtsson.  Probability measures associated to geodesics in the space of Kähler metrics.  In Algebraic and analytic microlocal analysis. AAMA, Evanston, Illinois, USA, May 14–26, 2012 and May 20–24, 2013. Contributions of the workshops, pages 395–419. Cham: Springer, 2018.
  * [7] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan.  Existence of minimal models for varieties of log general type.  J. Am. Math. Soc., 23(2):405–468, 2010.
  * [8] T. Bouche.  Sur les inégalités de Morse holomorphes lorsque la courbure du fibré en droites est dégénérée.  Ann. Sc. Norm. Super. Pisa, 18(4):501–523, 1991.
  * [9] S. Boucksom and D. Eriksson.  Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry, ArXiv: 1805.01016, to appear in Adv. Math.  2018\.
  * [10] S. Boucksom and M. Jonsson.  A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations, ArXiv: 2107.11221.  2021\.
  * [11] D. Catlin.  The Bergman kernel and a theorem of Tian.  In Proceedings of the 40th Taniguchi symposium, Katata, Japan, June 23–28, 1997, pages 1–23. Boston, MA: Birkhäuser, 1999.
  * [12] H. Chen.  Convergence of Harder-Narasimhan polygons.  Mém. Soc. Math. Fr., Nouv. Sér., 120:112, 2010.
  * [13] X. Chen.  The space of Kähler metrics.  J. Differ. Geom., 56(2):189–234, 2000.
  * [14] X. Chen and S. Sun.  Space of Kähler metrics. V: Kähler quantization.  In Metric and differential geometry. The Jeff Cheeger anniversary volume. Selected papers based on the presentations at the international conference on metric and differential geometry, Tianjin and Beijing, China, May 11–15, 2009, pages 19–41. Berlin: Springer, 2012.
  * [15] X. Dai, K. Liu, and X. Ma.  On the asymptotic expansion of bergman kernel.  J. Diff. Geom., 72(1):1–41, 2006.
  * [16] T. Darvas.  The Mabuchi geometry of finite energy classes.  Adv. Math., 285:182–219, 2015.
  * [17] T. Darvas.  Weak geodesic rays in the space of Kähler potentials and the class $\mathcal{E}(X,\omega)$.  J. Inst. Math. Jussieu, 16(4):837–858, 2017.
  * [18] T. Darvas, C. H. Lu, and Y. A. Rubinstein.  Quantization in geometric pluripotential theory.  Commun. Pure Appl. Math., 73(5):1100–1138, 2020.
  * [19] N. G. de Bruijn and P. Erdős.  Some linear and some quadratic recursion formulas. II.  Nederl. Akad. Wet., Proc., Ser. A, 55:152–163, 1952.
  * [20] J.-P. Demailly.  Estimations $L^{2}$ pour l’opérateur $\overline{\partial}$ d’un fibre vectoriel holomorphe semi-positif au-dessus d’une variété Kaehlerienne complete.  Ann. Sci. Éc. Norm. Supér. (4), 15:457–511, 1982.
  * [21] J.-P. Demailly.  Analytic methods in algebraic geometry.  Somerville, MA: International Press; Beijing: Higher Education Press, 2012.
  * [22] J.-P. Demailly.  Complex Analytic and Differential Geometry.  2012\.
  * [23] J.-P. Demailly.  Extension of holomorphic functions defined on non reduced analytic subvarieties.  In The legacy of Bernhard Riemann after one hundred and fifty years. Volume I, pages 191–222. Somerville, MA: International Press; Beijing: Higher Education Press, 2016.
  * [24] J.-P. Demailly, L. Ein, and R. Lazarsfeld.  A subadditivity property of multiplier ideals.  Mich. Math. J., 48:137–156, 2000.
  * [25] S. K. Donaldson.  Symmetric spaces, Kähler geometry and Hamiltonian dynamics.  In Northern California symplectic geometry seminar, pages 13–33. Providence, RI: American Mathematical Society, 1999.
  * [26] M. Fekete.  On the distribution of roots of algebraic equations with integral coefficients.  Math. Z., 17:228–249, 1923.
  * [27] S. Finski.  Semiclassical Ohsawa-Takegoshi extension theorem and asymptotic of the orthogonal Bergman kernel, arXiv: 2109.06851.  2021\.
  * [28] S. Finski.  Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions, arXiv:2201.04102.  2022\.
  * [29] S. Finski.  The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds, arXiv: 2207.02761.  2022\.
  * [30] O. Goldman and N. Iwahori.  The space of p-adic norms.  Acta Math., 109:137–177, 1963.
  * [31] W. E. Gryc and T. Kemp.  Duality in Segal-Bargmann spaces.  J. Funct. Anal., 261(6):1591–1623, 2011.
  * [32] W. E. Gryc and T. Kemp.  On sharp constants for dual Segal-Bargmann $L^{p}$ spaces.  J. Math. Anal. Appl., 424(2):1198–1222, 2015.
  * [33] S. Janson, J. Peetre, and R. Rochberg.  Hankel forms and the Fock space.  Rev. Mat. Iberoam., 3(1):61–138, 1987.
  * [34] L. Lami, C. Palazuelos, and A. Winter.  Ultimate data hiding in quantum mechanics and beyond.  Commun. Math. Phys., 361(2):661–708, 2018.
  * [35] L. Lempert.  The principle of least action in the space of Kähler potentials, arXiv:2009.09949.  2020\.
  * [36] L. Lempert.  On the Bergman kernels of holomorphic vector bundles, arXiv: 2109.08593.  2021\.
  * [37] Z. Lu.  On the Lower Order Terms of the Asymptotic Expansion of Tian-Yau-Zelditch.  Amer. J. Math., 122(2):235–273, 2000.
  * [38] X. Ma and G. Marinescu.  Holomorphic Morse inequalities and Bergman kernels, volume 254 of Progr. Math. Birkhäuser Verlag Basel, 2007.
  * [39] X. Ma and G. Marinescu.  Exponential estimate for the asymptotics of Bergman kernels.  Math. Ann., 362(3-4):1327–1347, 2015.
  * [40] T. Mabuchi.  Some symplectic geometry on compact Kähler manifolds. I.  Osaka J. Math., 24:227–252, 1987.
  * [41] T. V. Nguyen and A. Zeriahi.  Familles de polynômes presque partout bornees.  Bull. Sci. Math., II. Sér., 107:81–91, 1983.
  * [42] D. W. Nyström.  Test configurations and Okounkov bodies.  Compos. Math., 148(6):1736–1756, 2012.
  * [43] T. Ohsawa.  On the extension of $L^{2}$ holomorphic functions. V: Effects of generalization.  Nagoya Math. J., 161:1–21 (2001); erratum ibid. 163, 229, 2001\.
  * [44] D. H. Phong and J. Sturm.  The Monge-Ampère operator and geodesics in the space of Kähler potentials.  Invent. Math., 166(1):125–149, 2006.
  * [45] G. Pisier.  Un théorème sur les opérateurs linéaires entre espaces de Banach qui se factorisent par un espace de Hilbert.  Ann. Sci. Éc. Norm. Supér. (4), 13:23–43, 1980.
  * [46] H. Randriambololona.  Hauteurs pour les sous-schémas et exemples d’utilisation de méthodes arakeloviennes en théorie de l’approximation diophantienne (thèse).  2002\.
  * [47] H. Randriambololona.  Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faiseaux cohérents.  J. Reine Angew. Math., 590:67–88, 2006.
  * [48] R. Reboulet.  The asymptotic Fubini-Study operator over general non-Archimedean fields.  Math. Z., 299(3-4):2341–2378, 2021.
  * [49] R. A. Ryan.  Introduction to tensor products of Banach spaces.  Springer Monogr. Math. London: Springer, 2002.
  * [50] S. Semmes.  Complex Monge-Ampère and symplectic manifolds.  Am. J. Math., 114(3):495–550, 1992.
  * [51] J. Siciak.  Families of polynomials and determining measures.  Ann. Fac. Sci. Toulouse, Math. (5), 9(2):193–211, 1988.
  * [52] C. D. Sogge.  Fourier integrals in classical analysis, volume 105.  Cambridge: Cambridge University Press, 1993.
  * [53] J. Sun.  On the image of the Hilbert map, arXiv: 2208.13407.  2022\.
  * [54] G. Székelyhidi.  Filtrations and test-configurations.  Math. Ann., 362(1-2):451–484, 2015.
  * [55] G. Tian.  On a set of polarized Kähler metrics on algebraic manifolds.  J. Diff. Geom., 32(1):99–130, 1990.
  * [56] X. Wang.  Canonical metrics on stable vector bundles.  Commun. Anal. Geom., 13(2):253–285, 2005.
  * [57] S. Zelditch.  Szegö kernels and a theorem of Tian.  Internat. Math. Res. Notices, 1998(6):317–331, 1998.
  * [58] K. Zhu.  Analysis on Fock spaces, volume 263 of Grad. Texts Math. New York, NY: Springer, 2012.

Siarhei Finski, CNRS-CMLS, École Polytechnique F-91128 Palaiseau Cedex,
France.

E-mail<EMAIL_ADDRESS>


#  LADA: Look-Ahead Data Acquisition via Augmentation for Active Learning

Yoon-Yeong Kim, Kyungwoo Song, JoonHo Jang, Il-Chul Moon

###### Abstract

Active learning effectively collects data instances for training deep learning
models when the labeled dataset is limited and the annotation cost is high.
Besides active learning, data augmentation is also an effective technique to
enlarge the limited amount of labeled instances. However, the potential gain
from virtual instances generated by data augmentation has not been considered
in the acquisition process of active learning yet. Looking ahead the effect of
data augmentation in the process of acquisition would select and generate the
data instances that are informative for training the model. Hence, this paper
proposes Look-Ahead Data Acquisition via augmentation, or LADA, to integrate
data acquisition and data augmentation. LADA considers both 1) unlabeled data
instance to be selected and 2) virtual data instance to be generated by data
augmentation, in advance of the acquisition process. Moreover, to enhance the
informativeness of the virtual data instances, LADA optimizes the data
augmentation policy to maximize the predictive acquisition score, resulting in
the proposal of InfoMixup and InfoSTN. As LADA is a generalizable framework,
we experiment with the various combinations of acquisition and augmentation
methods. The performance of LADA shows a significant improvement over the
recent augmentation and acquisition baselines which were independently applied
to the benchmark datasets.

## Introduction

Large-scale datasets in the big data era have opened the blooming of
artificial intelligence, but the data labeling requires significant efforts
from human annotators. Therefore, an adaptive sampling, i.e. Active Learning,
has been developed to select the most informative data instances in learning
the decision boundary (Cohn, Ghahramani, and Jordan 1996; Tong 2001; Settles
2009). This selection is difficult because it is influenced by the learner and
the dataset at the same time. Hence, the understanding of the relation between
the learner and the dataset has become the components of active learning,
which queries the next training example by the informativeness for learning
the decision boundary.

(a) Max Entropy

(b) Mixup applied
after Max Entropy

(c) LADA without
learnable augmentation

(d) LADA with
learnable augmentation

Figure 1: Illustration of different scenarios for applying acquisition and
augmentation; with selected instances ($\star$), virtual instances ($\times$),
current decision boundary (solid line), and updated decision boundary (dashed
line). (a) Max Entropy selects the instances near the decision boundary and
updates the boundary. (b) If we augment the selected instances afterward, the
virtual instances may not be useful for updating the boundary. (c) By
considering the potential gain of the virtual instances from augmentation in
advance of the acquisition, we enhance the informativeness of both the
selected and augmented instances for learning the decision boundary. (d)
Moreover, we train the augmentation policy to maximize the acquisition score
of the virtual instances to generate useful instances.

Besides active learning, data augmentation is another source of providing
virtual data instances to train models. The labeled data may not cover the
full variation of the generalized data instances, so the learner has used the
data augmentation; particularly in the vision community (Liu and Ferrari 2017;
Perez and Wang 2017; Cubuk et al. 2019). Conventional data augmentation has
been a simple transformation of labeled data instances, i.e. flipping,
rotating, etc. Recently, the data augmentation has expanded to become a deep
generative model, such as Generative Adversarial Networks (GAN) (Goodfellow et
al. 2014) or Variational Autoencoder (VAE) (Kingma and Welling 2014), that
generate virtual examples. Since the conventional augmentations and the
generative model-based augmentations perform the Vicinal Risk Minimization
(VRM)(Chapelle et al. 2001), they assume that the virtual data instances in
the vicinity share the same label, which leads to limiting the feasible
vicinity. To overcome the limited vicinity of VRM, Mixup and its variants have
been proposed by interpolating multiple data instances (Zhang et al. 2017).
The pair of interpolated features and labels, or the Mixup instance, becomes a
virtual instance to enlarge the support of the training distribution.

Data augmentation and active learning intend to overcome the scarcity of
labeled dataset in different directions. First, active learning emphasizes the
optimized selection of the unlabeled real-world instances for the oracle
query, so there is no consideration on the benefit of the virtual data
generation. Second, the data augmentation focuses on generating an informative
virtual data instance without intervening on the data selection stage, and
without the potential assistance from oracle. These differences motivate us to
propose the Look-Ahead Data Acquisition via augmentation, or LADA framework.

LADA looks ahead the effect of data augmentation in advance of the acquisition
process, and LADA selects data instances by considering both unlabeled data
instances and virtual data instances generated by data augmentation, at the
same time. Whereas the conventional acquisition function does not consider the
potential gain of the data augmentation, LADA contemplates the informativeness
of the virtual data instances by integrating data augmentation into the
acquisition process. Figure 1 describes the different behavior of LADA and
conventional acquisition functions when applying data augmentation to active
learning.

Here are our contributions from the methodological and the experimental
perspectives. First, we propose a generalized framework, named LADA, that
looks ahead the acquisition score of the virtual data instance to be
augmented, in advance of the acquisition. Second, we train the data
augmentation policy to maximize the acquisition score to generate informative
virtual instances. Particularly, we propose two data augmentation methods,
InfoMixup and InfoSTN, which are trained by the feedback of acquisition
scores. Third, we substantiate the proposed framework by implementing the
variations of acquisition-augmentation frameworks with known acquisitions and
augmentation methods.

## Preliminaries

### Problem Formulation

This paper trains a classifier network, $f_{\theta}$, with dataset
$\mathscr{X}$ while our scenario is differentiated by assuming
$\mathscr{X}=\mathscr{X}_{U}\cup\mathscr{X}_{L}$ and
$|\mathscr{X}_{U}|\gg|\mathscr{X}_{L}|$. Here, $\mathscr{X}_{U}$ is a set of
unlabeled data instances, and $\mathscr{X}_{L}$ is a labeled dataset. Given
these notations, a data augmentation function,
$f_{aug}(x;\tau)\colon\mathscr{X}\to V(\mathscr{X})$, transforms a data,
$x\in\mathscr{X}$, into a modified data, $\tilde{x}\in V(\mathscr{X})$; where
$\tau$ is a parameter describing the policy of transformation, and
$V(\mathscr{X})$ is the vicinity set of $\mathscr{X}$ (Chapelle et al. 2001).
On the other hand, a data acquisition function,
$f_{acq}(x;f_{\theta})\colon\mathscr{X}_{U}\to\mathbb{R}$, calculates a score
of each data instance, $x\in\mathscr{X}_{U}$, based on the current classifier,
$f_{\theta}$; and $f_{acq}$ represents the instance selection strategy in the
learning procedure of $f_{\theta}$ with the instance, $x\in\mathscr{X}_{U}$.

### Data Augmentation

In the conventional data augmentations, $\tau$ in $f_{aug}(x;\tau)$ indicates
the predefined degree of rotating, flipping, cropping, etc. $\tau$ is manually
chosen by the modeler to describe the vicinity of each data instance.

Another approach of modeling $\tau$ is utilizing the feedback from the current
classifier network, $f_{\theta}$. Spatial Transformer Network (STN) is a
transformer to generate a virtual example by training $\tau$ to minimize the
cross-entropy (CE) loss of the transformed data (Jaderberg et al. 2015):

$\tau^{*}=\operatorname*{argmin}_{\tau}CE(f_{\theta}(f^{STN}_{aug}(x;\tau)),y),$
(1)

where $y$ is the ground-truth label of the data instance, $x$.

Recently, Mixup-based data augmentations generate a virtual data instance from
the vicinity of a pair of data instances. In Mixup, $\tau$ becomes the mixing
policy of two data instances, $x_{i}$ and $x_{j}$ (Zhang et al. 2017):

$f^{Mixup}_{aug}(x_{i},x_{j};\tau)=\lambda
x_{i}+(1-\lambda)x_{j},\lambda\sim\text{Beta}(\tau,\tau),$ (2)

where the labels are also mixed by the proportion $\lambda$. While Eq.(2)
corresponds to the input feature mixture, Manifold Mixup mixes the hidden
feature maps from the middle of neural networks to learn smoother decision
boundary at multiple levels of representations (Verma et al. 2018). Whereas
$\tau$ is a fixed value without any learning process, AdaMixup learns $\tau$
by adopting a discriminator, $\varphi^{ada}$ (Guo, Mao, and Zhang 2019):

$\displaystyle\tau^{*}$
$\displaystyle=\operatorname*{argmax}_{\tau}\log\text{P}(\varphi^{ada}(f^{Mixup}_{aug}(x_{i},x_{j};\tau))=1)$
$\displaystyle+\log\text{P}(\varphi^{ada}(x_{i})=0)+\log\text{P}(\varphi^{ada}(x_{j})=0).$
(3)

### Active Learning

We focus on the pool-based active learning with uncertainty score (Settles
2009). Given this scope of active learning, the data acquisition function
measures the utility score of the unlabeled data instances, i.e.
$x^{*}=\operatorname*{argmax}_{x}f_{acq}(x;f_{\theta}).$

The traditional acquisition functions measure the predictive entropy,
$f_{acq}^{Ent}(x;f_{\theta})=\mathbb{H}[y|x;f_{\theta}]$ (Shannon 1948); or
the variation ratio,
$f_{acq}^{Var}(x;f_{\theta})=1-max_{y}\text{P}(y|x;f_{\theta})$ (Freeman
1965). The recent acquisition function calculates the hypothetical
disagreement by $f_{\theta}$ on a data instance,
$f_{acq}^{BALD}(x;f_{\theta})=\mathbb{H}[y|x;f_{\theta}]-\mathbb{E}_{\text{P}(\theta|D_{train})}[\mathbb{H}[y|x;f_{\theta}]]$
(Houlsby et al. 2011).

Besides the classifier network, $f_{\theta}$, additional modules are applied
to measure the acquisition score. To find the most dissimilar instance in
$\mathscr{X}_{U}$ compared to $\mathscr{X}_{L}$, a discriminator,
$\varphi^{VAAL}$, is introduced to estimate the probability of belonging to
$\mathscr{X}_{U}$ (Sinha, Ebrahimi, and Darrell 2019):

$\displaystyle
f_{acq}^{VAAL}(x;\varphi^{VAAL})=\text{P}(x\in\mathscr{X}_{U};\varphi^{VAAL}).$
(4)

To diversely select uncertain data instances, the gradient embedding from the
pseudo label, $\hat{y}$, is used in k-MEANS++ seeding algorithm (Ash et al.
2020):

$\displaystyle
f_{acq}^{BADGE}(x;f_{\theta})=\frac{\partial}{\partial\theta_{out}}CE(f_{\theta}(x),\hat{y}).$
(5)

### Active Learning with Data Augmentation

There are a few prior works in integrating active learning and data
augmentation effectively. Bayesian Generative Active Deep Learning (BGADL)
integrates acquisition and augmentation by selecting data instances via
$f_{acq}$, then augmenting the selected instances via $f_{aug}$, which is VAE,
afterward (Tran et al. 2019). However, BGADL limits the vicinity to preserve
the labels, and BGADL demands on large labeled instances to train generative
models. More importantly, BGADL does not consider the potential gain of data
augmentation in the process of acquisition.

(a) Overview of LADA framework

(b) Training process of the policy generator network, $\pi_{\phi}$,
in LADA with Max Entropy and Manifold Mixup

Figure 2: Overall algorithm of LADA and training process of data augmentation.
(a) $f_{acq}$ considers both the unlabeled instances and the virtual instances
generated from $f_{aug}$. Moreover, $f_{aug}$ is optimized with the feedback
of $f_{acq}$. (b) The parameters of the classifier network $f_{\theta}$ are
fixed during the backpropagation for the policy generator network.

## Methodology

A contribution of this paper is proposing an integrated framework of data
augmentation and acquisition, so we start from formulating such a framework.
Afterward, we propose an integrated function for acquisition and augmentation
as an example of the implemented framework.

### Look-Ahead Data Acquisition via Augmentation

Since we look ahead the acquisition score of augmented data instances, it is
natural to integrate the functionalities of acquisitions and augmentations.
This paper proposes a Look-Ahead Data Acquisition via augmentation, or LADA
framework. Figure 2(a) depicts the LADA framework, which consists of the data
augmentation component and the acquisition component. The goal of LADA is
enhancing the informativeness of both 1) real-world data instance, which is
unlabeled at current, but will be labeled by the oracle in the future; and 2)
virtual data instance, which will be generated from the unlabeled data
instances that are selected. This goal is achieved by looking ahead of their
acquisition scores before actual selections for the oracle annotations.

Specifically, LADA trains the data augmentation function, $f_{aug}(x;\tau)$,
to maximize the acquisition score of the transformed data instance of $x_{U}$
before the oracle annotations. Eq.(6) specifies the learning objectives of the
augmentation policy via the feedback from acquisition.

$\tau^{*}=\operatorname*{argmax}_{\tau}f_{acq}(f_{aug}(x_{U};\tau);f_{\theta}).$
(6)

With the optimal $\tau^{*}$ corresponding to $x_{U}$, $f_{acq}$ calculates the
acquisition score of $x_{U}$ (see Eq.(7)), and the score also considers the
utility of the augmented instance, $f_{aug}(x_{U};\tau^{*})$:

$x_{U}^{*}=\operatorname*{argmax}_{x_{U}\in\mathscr{X}_{U}}[f_{acq}(x_{U};f_{\theta})+f_{acq}(f_{aug}(x_{U};\tau^{*});f_{\theta})]$
(7)

Whereas the proposed LADA framework is a generalized framework that can adopt
the various types of acquisition and augmentation functions, this section
mainly adopts Mixup for $f_{aug}$, i.e. $f_{aug}^{Mixup}$ and Max Entropy for
$f_{acq}$, i.e. $f_{acq}^{Ent}$. To begin with, we introduce an integrated
single function to substitute the composition of functions as
$f_{integ}=f_{acq}\circ
f_{aug}(x_{U})=f_{acq}(f_{aug}(x_{U};\tau);f_{\theta})$ for generality.

### Integrated Augmentation and Acquisition: InfoMixup

As we introduce LADA with $f_{integ}$ to look ahead the acquisition score of
the virtual data instances, $f_{integ}$ can be a simple composition of well-
known acquisition functions and augmentation functions where the policy of
augmentation is fixed. However, this does not enhance the informativeness of
the virtual data instances. Hence, we propose the integration where the policy
of data augmentation is optimized to maximize the acquisition score, within a
single function. Here, we introduce InfoMixup as a learnable data
augmentation.

#### Data Augmentation

First, we propose InfoMixup, which is an adaptive version of Mixup to
integrate the data augmentation into active learning. InfoMixup learns its
mixing policy, $\tau$, by the objective functions Eq.(8), where
$\lambda\sim\text{Beta}(\tau^{*},\tau^{*})$ maximizes the acquisition score of
the virtual data instance resulting from mixing two randomly paired data
instances, $x_{i}$ and $x^{\prime}_{i}$:

$\displaystyle
f_{EntMix}(x_{i},x^{\prime}_{i};\tau,f_{\theta})=f_{acq}^{Ent}(f_{aug}^{Mixup}(x_{i},x^{\prime}_{i};\tau);f_{\theta})$
$\displaystyle\tau^{*}=\operatorname*{argmax}_{\tau}f_{EntMix}(x_{i},x^{\prime}_{i};\tau_{,}f_{\theta}).$
(8)

InfoMixup is the starting ground where we correlate the data augmentation
guided by the data acquisition from the perspective of the predictive
classifier entropy.

We adopt Manifold Mixup as the data augmentation at the hidden layer.
Specifically, the pair of $(x_{i},x^{\prime}_{i})\in\mathscr{X}_{U}$ is
processed through the current classifier network, $f_{\theta}$, until the
propagation reaches the randomly selected $k$-th layer111Throughout this
paper, we denote the forward path from the $i^{th}$ layer to the $j^{th}$
layer of the classifier network as $f_{\theta}^{i:j}$, where $0$-th is the
input layer and $L$-th is the output layer. Hence,
$f_{\theta}=f_{\theta}^{0:L}$.. Afterwards, the $k$-th feature maps
$(h^{k}(x_{i}),h^{k}(x^{\prime}_{i}))$ are concatenated and processed by the
policy generator network, $\pi_{\phi}$, to predict $\tau_{i}^{*}$ that
maximizes the acquisition score.

#### Data Augmentation Policy Learning

As we formulate the Mixup based augmentation, we propose a policy generator
network, $\pi_{\phi}$, to perform the amortized inference on the Beta
distribution of InfoMixup. While we provide the details of the policy network
in Appendix A.2 and Figure 2(b), we formulate this inference process as Eq.(9)
and Eq.(10).

$\displaystyle h^{k}(x_{i})$
$\displaystyle=f_{\theta}^{0:k}(x_{i}),h^{k}(x^{\prime}_{i})=f_{\theta}^{0:k}(x^{\prime}_{i})$
(9) $\displaystyle\tau_{i}$
$\displaystyle=\pi_{\phi}([h^{k}(x_{i}),h^{k}(x^{\prime}_{i})])$ (10)
$\displaystyle=MLP_{\phi}([h^{k}(x_{i}),h^{k}(x^{\prime}_{i})]).$

To train the parameters, $\phi$, of the policy generator network, $\pi$, the
paired features are mixed-up with $N$ sampled $\lambda_{i}$’s. Using a $n$-th
sampled $\lambda_{i}^{n}$ from the Beta distribution inferred by $\pi_{\phi}$,
the feature maps $h^{k}(x_{i})$ and $h^{k}(x^{\prime}_{i})$ are mixed to
produce $h^{k,n}_{m}(x_{i},x^{\prime}_{i})$ as the below:

$\displaystyle\lambda_{i}^{n}$
$\displaystyle\sim\text{Beta}(\tau_{i},\tau_{i}),$ (11) $\displaystyle
h^{k,n}_{m}(x_{i},x^{\prime}_{i})$
$\displaystyle=\lambda_{i}^{n}h^{k}(x_{i})+(1-\lambda_{i}^{n})h^{k}(x^{\prime}_{i}).$
(12)

By processing $h^{k,n}_{m}$ for the rest layers of the classifier network, the
predictive class probability of the mixed features is obtained as
$\hat{y}_{i}^{n}=f_{\theta}^{k:L}(h^{k,n}_{m}(x_{i},x^{\prime}_{i}))$. In
order to generate a useful virtual instance through InfoMixup, the policy
generator network has a loss function to minimize the negative value of the
predictive entropy as Eq.(13), and the predictive entropy is a component of
$f_{acq}$, which provides the incentive for the integration of acquisition and
augmentation. The gradient of this loss function is calculated by averaging
the $N$ entropy values of the replicated mixed features. It should be noted
that Eq.(13) embed $\phi$ in the generation of
$h^{k,n}_{m}(x_{i},x^{\prime}_{i})$, so the gradient can be estimated via the
Monte-Carlo sampling (Hastings 1970). Figure 2(b) illustrates the forward and
the backward paths for the training process of the policy generator network.

$\displaystyle\frac{\partial}{\partial\phi}L_{\pi}=\frac{\partial}{\partial\phi}$
$\displaystyle(-{1\over
N}\sum_{n}^{N}f_{acq}(h^{k,n}_{m}(x_{i},x^{\prime}_{i});f_{\theta}^{k:L}))$
(13) $\displaystyle=\frac{\partial}{\partial\phi}$ $\displaystyle(-{1\over
N}\sum_{n}^{N}\mathbb{H}[\hat{y}_{i}^{n}|h^{k,n}_{m}(x_{i},x^{\prime}_{i});f_{\theta}^{k:L}])$

In the backpropagation, we have a process of sampling $\lambda_{i}$s from the
Beta distribution parameterized by $\tau_{i}$. To enable the backpropagation
signals to pass by, we follow the reparameterization technique of the optimal
mass transport (OMT) gradient estimator, which utilizes the implicit
differentiation (Jankowiak and Obermeyer 2018; Jankowiak and Karaletsos 2019).
Appendix B provides the details of our OMT gradient estimator in the
backpropagation process.

#### Data Acquisition by Learned Policy

After optimizing the mixing policy, $\tau_{i}^{*}$, for $i$-th pair of
unlabeled data instances, $(x_{i},x^{\prime}_{i})$, we calculate the joint
acquisition score of the data pair by aggregating the individual acquisition
scores of 1) $x_{i}$, 2) $x^{\prime}_{i}$, and 3) their mixed feature maps,
$h_{m}^{k,n}(x_{i},x^{\prime}_{i})$ as the below:

$\displaystyle\hat{y}_{i}=f_{\theta}(x_{i}),\hat{y}^{\prime}_{i}=f_{\theta}$
$\displaystyle(x^{\prime}_{i}),\hat{y}_{i}^{n}=f_{\theta}^{k:L}(h_{m}^{k,n}(x_{i},x^{\prime}_{i}))$
(14) $\displaystyle f_{acq}\big{(}(x_{i},x^{\prime}_{i});f_{\theta}\big{)}$
$\displaystyle=\mathbb{H}[\hat{y}_{i}|x_{i};f_{\theta}]+\mathbb{H}[\hat{y^{\prime}}_{i}|x^{\prime}_{i};f_{\theta}]$
(15) $\displaystyle+{1\over
N}\sum_{n}^{N}\mathbb{H}[\hat{y}_{i}^{n}|h^{k,n}_{m}(x_{i},x^{\prime}_{i});f_{\theta}^{k:L}]$

As we calculate the acquisition score by including the predictive entropy of
the InfoMixup feature map, the acquisition is influenced by the data
augmentation. More importantly, this integration is reciprocal because the
optimal augmentation policy of InfoMixup comes from the acquisition score.
This reciprocal relation is an example of motivating the LADA framework by
overcoming the separation between the augmentation and the acquisition.

If we take InfoMixup as an example of LADA, we show that InfoMixup generates a
virtual sample with a high predictive entropy in the class estimation, which
could be regarded as a decision boundary region that is not clearly explored,
yet. The unexplored region is identified by the optimal policy of $\tau^{*}$
in the acquisition.

Here, we introduce a pipelined variant in $f_{integ}$ to emphasize the worth
of the integration. One possible variation is incorporating Mixup-based data
augmentation and acquisition function as a two-step model, where 1) the
acquisition function selects the data instances whose individual scores are
the highest, 2) then Mixup is afterward applied to the selected instances.
However, this method may increase the criteria of individual data instances
laid in the first and the second terms of Eq.(15), but it may not optimize the
criteria of their mixing process laid in the last part of Eq.(15) since it has
not considered the effect of Mixup in the selection process. This may not
enhance the informativeness of the virtual data instances. We compare this
variation with LADA in the Experiments Section.

Algorithm 1 LADA with Max Entropy and Manifold Mixup

1:Labeled dataset $\mathscr{X}_{L}^{0}$, Classifier $f_{\theta}$

2:for $j=0,1,2,\ldots$ do $\triangleright$ active learning

3: Randomly sample $\mathscr{X}_{U}^{pool}\subset\mathscr{X}_{U}$

4: Get $\mathscr{X^{\prime}}_{U}^{pool}$ which is randomly shuffled
$\mathscr{X}_{U}^{pool}$

5: Randomly chose the layer index $k$ of the $f_{\theta}$

6: for $i=0,1,2,\ldots$ do$\triangleright$ for the unlabeled instances

7: $x_{i}\in\mathscr{X}_{U}^{pool}$,
$x^{\prime}_{i}\in\mathscr{X^{\prime}}_{U}^{pool}$

8: Get $h^{k}(x_{i}),h^{k}(x^{\prime}_{i})$ of $x_{i},x^{\prime}_{i}$ as
Eq.(9)

9: for $q=0,1,2,\ldots$ do $\triangleright$ training of $\pi_{\phi}$

10: $\tau^{q}_{i}=\pi_{\phi}([h^{k}(x_{i});h^{k}(x^{\prime}_{i})])$

11: Calculate $\frac{\partial}{\partial\phi}L_{\pi}$ as Eq.(13)

12: $\phi\leftarrow\phi-\eta_{\pi}\frac{\partial}{\partial\phi}L_{\pi}$

13: if $L_{\pi}$ is minimal then

14: $\tau^{*}_{i}=\tau^{q}_{i}$

15: end if

16: end for

17: end for

18: Select and query the dataset, $\mathscr{X}_{S}$, as Eq.(16)

19: $\mathscr{X}_{L}^{j+1}=\mathscr{X}_{L}^{j}\cup\mathscr{X}_{S}$

20: for $t=0,1,2,\ldots$ do $\triangleright$ training of $f_{\theta}$

21: $\lambda_{i}\sim\text{Beta}(\tau_{i}^{*},\tau_{i}^{*})$ for
$(x_{i},x^{\prime}_{i})\in\mathscr{X}_{S}$

22: Get virtual dataset, $\mathscr{X}_{M}$, as Eq.(17)

23: Calculate $L_{f}$ as Eq.(18) $\sim$ (19)

24: $\theta\leftarrow\theta-\eta_{f}\frac{\partial}{\partial\theta}L_{f}$

25: end for

26:end for

### Training Set Expansion through Acquisition

We assume that we start the $j^{th}$ active learning iteration with already
acquired labeled dataset $\mathscr{X}_{L}^{j}$. With the allowed budget per
acquisition as $k$, we acquire top-$\frac{k}{2}$ pairs, i.e. $\mathscr{X}_{S}$
among the subsets,
$\mathscr{X}^{\prime}_{S}\subset\mathscr{X}_{U}\times\mathscr{X}_{U}$, with
$|\mathscr{X}^{\prime}_{S}|=\frac{k}{2}$.

$\mathscr{X}_{S}=\operatorname*{argmax}_{\mathscr{X}^{\prime}_{S}\subset\mathscr{X}_{U}\times\mathscr{X}_{U}}\sum_{(x_{i},x^{\prime}_{i})\in\mathscr{X}^{\prime}_{S}}f_{acq}((x_{i},x^{\prime}_{i});f_{\theta})$
(16)

At this moment, oracle annotates the true labels on $\mathscr{X}_{S}$. Also,
we have a virtual instance dataset, $\mathscr{X}_{M}$, generated by InfoMixup
with the optimal mixing policy, $\tau^{*}$:

$\mathscr{X}_{M}=\bigcup_{(x_{i},x^{\prime}_{i})\in\mathscr{X}_{S}}\\{\lambda_{i}f_{\theta}^{0:L}(x_{i})+(1-\lambda_{i})f_{\theta}^{0:k}(x^{\prime}_{i})\\},$
(17)

where $\lambda_{i}\sim\text{Beta}(\tau_{i}^{*},\tau_{i}^{*})$. Here,
$\tau^{*}$ is dynamically inferred by the neural network of $\pi_{\phi}$ per
each pair.

Up to this phase, our training dataset becomes
$\mathscr{X}_{L}^{j+1}=\mathscr{X}_{L}^{j}\cup\mathscr{X}_{S}$ and
$\mathscr{X}_{M}$. Our proposed algorithm, described in Algorithm 1, utilizes
$\mathscr{X}_{M}$ for this active learning iteration only, with various
$\lambda_{i}$s sampled at each training epoch. The classifier network’s
parameter, $\theta$, is learned via the gradient of the cross-entropy loss,

$\displaystyle L_{f}$
$\displaystyle=CE(f_{\theta}(x_{i}),y_{i})_{x_{i}\in\mathscr{X}_{L}^{j+1}}$
(18)
$\displaystyle+CE(f_{\theta}^{k:L}(x_{i}),y_{i})_{x_{i}\in\mathscr{X}_{M}},$
(19)

where $y_{i}$ is the corresponding ground-truth label annotated from the
oracle for Eq.(18); or the mixed label according to the mixing policy for
Eq.(19).

###  LADA with Various Augmentation-Acquisition

Since we propose the integrated framework of acquisition function and data
augmentation to look ahead the informativeness of the data, we can use various
acquisition functions and data augmentations in our LADA framework. For
example, we may substitute the Max Entropy, which is the feedback of the
acquisition function to the data augmentation in InfoMixup, with another
simple feedback, Var Ratio. Also, if we apply the VAAL acquisition function,
LADA with VAAL trains the generator network, $\pi_{\phi}$, to maximize the
discriminator’s indication on the unlabeled dataset,
$\text{P}(x\in\mathscr{X}_{U};\varphi^{VAAL})$, for the generated instances.

Similarly, we may substitute the data augmentation of InfoMixup with Spatially
Transform Networks (STN) (Jaderberg et al. 2015), a.k.a. InfoSTN. STN may be
trained with a subset of unlabeled data as input to maximize their predictive
entropy when propagated to the current classifier network. The score to pick
the most informative data is formulated as
$f_{acq}^{STN}(x_{i};f_{\theta})=\mathbb{H}[\hat{y}^{STN}_{i}|x^{STN}_{i}]$,
where $x^{STN}_{i}$ is the spatially transformed output of the data $x_{i}$,
and $\hat{y}^{STN}_{i}$ is the corresponding prediction by the current
classifier network. We provide more details in Appendix C.

## Experiments

### Baselines and Datasets

This section denotes the proposed framework as LADA, and we specify the
instantiated data augmentation and acquisition by its subscript, i.e. the
proposed InfoMixup as $\mathrm{LADA_{EntMix}}$ which adopts Max Entropy as
data acquisition and Mixup as data augmentation to select and generate
informative samples. If we change the entropy measure to the Var Ratio or the
discriminator logits of VAAL, it results in the subscript of VarMix or
VaalMix, respectively. Also, if we change the augmentation policy to the STN
network, the subscript becomes EntSTN.

We compare our models to 1) Coreset (Sener and Savarese 2018); 2) BADGE (Ash
et al. 2020); and 3) VAAL (Sinha, Ebrahimi, and Darrell 2019) as the baselines
for active learning. We also include some data augmented active learning: 1)
BGADL, 2) Manifold Mixup; and 3) AdaMixup. Here, BGADL is an integrated data
augmentation and acquisition method, but it should be noted that BGADL has no
learning mechanism in the augmentation from the feedback of acquisition. We
also add ablated baselines to see the effect of learning $\tau$, so we
introduce the fixed $\tau$ case as $\mathrm{LADA^{fixed}}$. The classifier
network, $f_{\theta}$, adopts Resnet-18 (He et al. 2016), and the policy
generator network, $\pi_{\phi}$, consists of a much smaller neural network.
Appendix A.2 provides more details on the networks and their training.

We experiment the above-mentioned models with three benchmark datasets:
FashionMNIST (Fashion) (Xiao, Rasul, and Vollgraf 2017), SVHN (Netzer et al.
2011), and CIFAR-10 (Krizhevsky, Hinton et al. 2009). Throughout our
experiments, we repeat the experiments for five times to validate the
statistical significance, and the maximum acquisition iteration is limited to
100. More details about the treatment on each dataset are in Appendix A.1.

We evaluate the models under the pool-based active learning scenario. We
assume that the model has 20 training instances, which are randomly chosen and
balanced. As the active learning iteration progresses, we acquire 10
additional training instances at each iteration, and we use the same amount of
oracle queries for all models, which results in selecting top-5 pairs when
adopting Mixup as data augmentation in the LADA framework.

Method | Fashion | SVHN | CIFAR-10 | Time | Param.
---|---|---|---|---|---
Baselines | Random | 80.96$\pm$0.62 | 73.92$\pm$2.80 | 35.27$\pm$1.36 | 1 | -
BALD | 80.99$\pm$0.59 | 75.66$\pm$2.07 | 34.71$\pm$2.28 | 1.36 | -
Coreset | 78.47$\pm$0.30 | 68.57$\pm$3.13 | 28.25$\pm$0.89 | 1.54 | -
BADGE | 80.94$\pm$0.98 | 70.89$\pm$1.91 | 28.60$\pm$1.17 | 1.31 | -
BGADL | 78.42$\pm$1.05 | 63.50$\pm$1.56 | 35.08$\pm$2.20 | 4.69 | 13M
Entropy-based | Max Entropy | 80.93$\pm$1.85 | 72.57$\pm$0.76 | 34.97$\pm$0.71 | 1.01 | -
Ent w.ManifoldMixup | 82.31$\pm$0.38 | 72.69$\pm$1.29 | 35.88$\pm$0.85 | 1.03 | -
Ent w.AdaMixup | 81.30$\pm$0.83 | 73.00$\pm$0.39 | 35.67$\pm$1.75 | 1.03 | 5K
$\mathrm{LADA^{fixed}_{EntMix}}$ | 83.08$\pm$1.34 | 75.73$\pm$1.48 | 36.34$\pm$0.88 | 1.06 | -
$\mathrm{LADA_{EntMix}}$ | 83.67$\pm$0.29 | 76.55$\pm$0.31 | 37.04$\pm$1.34 | 1.32 | 77K
$\mathrm{LADA^{fixed}_{EntSTN}}$ | 82.37$\pm$0.58 | 72.08$\pm$1.67 | 35.55$\pm$1.34 | 1.02 | 5K
$\mathrm{LADA_{EntSTN}}$ | 81.83$\pm$0.55 | 73.80$\pm$0.81 | 36.18$\pm$0.69 | 1.20 | 5K
VAAL -based | VAAL | 82.67$\pm$0.29 | 75.01$\pm$0.66 | 39.82$\pm$0.86 | 3.55 | 301K
$\mathrm{LADA^{fixed}_{VaalMix}}$ | 82.63$\pm$0.29 | 76.83$\pm$1.05 | 44.42$\pm$2.12 | 3.56 | 301K
$\mathrm{LADA_{VaalMix}}$ | 82.60$\pm$0.49 | 77.92$\pm$0.51 | 44.56$\pm$1.40 | 3.60 | 378K
VarRatio -based | Var Ratio | 81.05$\pm$0.18 | 74.07$\pm$1.87 | 34.99$\pm$0.73 | 1.01 | -
$\mathrm{LADA^{fixed}_{VarMix}}$ | 83.11$\pm$0.66 | 76.01$\pm$2.64 | 35.98$\pm$1.68 | 1.06 | -
$\mathrm{LADA_{VarMix}}$ | 84.47$\pm$0.89 | 76.09$\pm$0.94 | 36.84$\pm$0.51 | 1.33 | 77K
Table 1: Comparison of test accuracy, the run-time of 1 iteration of
acquisition (Time), and the number of parameters (Param.). The best
performance in each category is indicated as boldface. The run-time is
calculated as the ratio to the Random acquisition. The number of parameters is
only reported for the auxiliary network, and - indicates no auxiliary network
in the method.

(a) Fashion

(b) SVHN

(c) CIFAR-10

Figure 3: Test accuracy over the acquisition iterations

### Quantitative Performance Evaluations

Table 3 shows the average test accuracy, and the accuracy of each replication
represents the best accuracy over the acquisition iterations. Since we
introduce the generalizable framework, Table 3 separates the performances by
the instantiated acquisition functions. The group of baselines does not have
any learning mechanism on the acquisition metric, and this group has the worst
performances. We suggest three acquisition functions to be adopted by our LADA
framework, which are 1) the predictive entropy by the classifier, 2) the
discriminator logits in VAAL, and 3) the classifier variation ratio. Given
that VAAL uses a discriminator and a generator, the VAAL-based model has more
parameters to optimize in terms of complexity, which provides an advantage in
a complex dataset, such as CIFAR-10.

When we examine the general performance gains across datasets, we find the
best performers as $\mathrm{LADA_{VarMix}}$ in Fashion; and
$\mathrm{LADA_{VaalMix}}$ in SVHN and CIFAR-10. In terms of the data
augmentation, Mixup-based augmentation outperforms STN augmentation. As the
dataset becomes complex, the greater gain of performance is achieved by
$\mathrm{LADA_{EntSTN}}$ in SVHN or CIFAR-10, compared to Fashion. In all
combinations of baselines and datasets, the integrations of augmentation and
acquisition, a.k.a. LADA variations, show the best performance in most cases.
In terms of the ablation study, the learning of the data augmentation policy,
$\tau$, is meaningful because the 10 learning case of LADA is better than the
fixed case in 12 variations of LADA. Figure 3 shows the convergence speed to
the best test accuracy by each model. As the dataset becomes complex, the
performance gain by LADA becomes apparent.

Additionally, we compare the integrated framework to the pipelined approach.
Max Entropy does not have an augmentation part, so it becomes the simplest
model. Then, Ent w.Manifold Mixup adds the Manifold Mixup augmentation, but it
does not have a learning process on the mixing policy. Finally, Ent w.AdaMixup
has a learning process on the mixing policy, but the learning is separated
from the acquisition. These pipelined approaches show lower performances than
the integration cases of LADA.

Finally, as LADA is a generalizable framework to work with the various
acquisition and augmentation functions, Figure 4(a) and Figure 4(b) show the
ablation study on the instantiated LADA with the VAAL acquisition function and
the $STN$ augmentation function, respectively. The figures confirm the effects
of both integration and learnable augmentation policy with feedback from the
acquisition.

(a) $\mathrm{LADA_{VaalMix}}$

(b) $\mathrm{LADA_{EntSTN}}$

Figure 4: Ablation study of LADA with various acquisition and augmentation on
CIFAR-10 dataset

### Qualitative Analysis on Acquired Data Instances

Besides the quantitative comparison, we need reasoning on the behavior of
LADA. Therefore, we selected $\mathrm{LADA_{EntMix}}$ to contrast to the
pipelined approach. We investigate on 1) achieving the informative data
instances by acquisition, 2) the validity of the optimal $\tau^{*}$ in the
augmentation learned from the policy generator network $\pi_{\phi}$, and 3)
examining the coverage of the explored space.

(a) Max Entropy at early

(b) Max Entropy at late

(c) $\mathrm{LADA_{EntMix}}$ at early

(d) $\mathrm{LADA_{EntMix}}$ at late

Figure 5: tSNE (Maaten and Hinton 2008) plot of acquired instance ($\star$)
and augmented instance ($\times$), with entropy values. The numbers written in
black indicate the predictive entropy of unlabeled data instances that were
selected from the unlabeled pool. The numbers written in red indicate the
maximum (*average) value of predictive entropy of the virtual data instances
that were generated from InfoMixup. The acquisition iterations of early and
late are 7 and 76.

To check the informativeness of data instances, Figure 5 shows the different
acquisition process between Max Entropy and $\mathrm{LADA_{EntMix}}$. Max
Entropy selects a data instance with the highest predictive entropy value.
Compared to Max Entropy, $\mathrm{LADA_{EntMix}}$ selects a pair of two data
instances with the highest value of the aggregated predictive entropy, which
is the summation of the predictive entropy from two data instances and one
InfoMixup instance. By mixing two unlabeled data instances with the
corresponding optimal mixing policy $\tau^{*}$, the virtual data instance,
generated along with the vicinal space, results in a high entropy value, which
can be higher than the selected instance by Max Entropy. The virtual data
instance helps the current classifier model to clarify the decision boundary
between two classes along the interpolation line of two mixed real instances.

To confirm the validity of the optimal $\tau^{*}$, we compare three cases of
1) the inferred $\tau$ ($\mathrm{LADA_{EntMix}}$); 2) the fixed $\tau$
($\mathrm{LADA^{fixed}_{EntMix}}$); and 3) the pipelined model’s $\tau$ (Ent
w.Manifold Mixup). Figure 6(a) shows the entropy of the virtual data instances
over the acquisition process. As expected, the optimal $\tau^{*}$ learned in
$\mathrm{LADA_{EntMix}}$ produces the highest entropy over the acquisition
process, but it should be noted that the differentiation becomes significant
after some acquisition iterations, which comes from the requirement of
training the classifier. Figure 6(b) shows the distribution of entropy of
virtual instances, with the median value of each interval as $x$-axis. This
also shows that the optimal $\tau^{*}$ has the highest density beyond the
interval of the median 2.2.

(a) Mean value of entropy
of the virtual data

(b) Number of virtual data
with entropy values

Figure 6: Entropy values of the virtual data generated from
$\mathrm{LADA_{EntMix}}$, $\mathrm{LADA^{fixed}_{EntMix}}$, and Ent w.Manifold
Mixup

To examine the coverage of the explored latent space, Figure 7 illustrates the
latent space of the acquired data instances and the augmented data instances.
Ent w.AdaMixup has a potential capability of interpolating distantly paired
data instances, but its learned $\tau$ limits a sample of $\lambda$ to be
placed near either one of the paired instances because of the aversion on the
manifold intrusion. Therefore, in the experiments, Ent w.AdaMixup ends up
exploring the space near the acquired instances. The generated virtual data
instances by $\mathrm{LADA_{EntMix}}$ show further exploration than Ent
w.AdaMixup. The latent space makes the linear interpolation of
$\mathrm{LADA_{EntMix}}$ to be curved by the manifold, but it keeps the
interpolation line of the curved manifold. The extent of the interpolation is
broader than AdaMixup because the optimal $\tau^{*}$ is guided by the entropy
maximization, which is adversarial in a sense. This adversarial approach is
different from the aversion of the manifold intrusion because the latter is
more conservative to the currently learned parameter.

(a) $\mathrm{LADA_{EntMix}}$

(b) Ent w.AdaMixup

Figure 7: tSNE plot of acquired data instances ($\star$) and generated virtual
data instances ($\times$). The labels are categorized by colors.

## Conclusions

In the real world where gathering a large-scale labeled dataset is difficult
because of the constrained human or computational resources, learning the deep
neural network requires an effective utilization of the limited resources.
This limitation motivated the integration of data augmentation and active
learning. This paper proposes a generalized framework for such integration,
named as LADA, which adaptively selects the informative data instances by
looking ahead the acquisition score of both 1) the unlabeled data instances
and 2) the virtual data instances to be generated by data augmentation, in
advance of the acquisition process. To enhance the effect of the data
augmentation, LADA learns the augmentation policy to maximize the acquisition
score. With repeated experiments on the various datasets and the comparison
models, LADA shows a considerable performance by selecting and augmenting
informative data instances. The qualitative analysis shows the different
behavior of LADA that finds the vicinal space of high acquisition score by
learning the optimal policy.

## Ethics Statement

In the real world, the limited amount of labeled dataset makes it hard to
train the deep neural networks and high cost of the annotation cost becomes
problematic. This leads to the decision on what to select and annotate first,
which calls upon the active learning. Besides the active learning, effectively
enlarging the limited amount of labeled dataset is also considerable. With
this motivations, we propose a framework that can adopt various types of
acquisitions and augmentations that exist in machine learning field. By
looking ahead the effect of data augmentation in the process of acquisition,
we can select data instances that are informative if selected and labeled but
also augmented. Moreover, by learning the augmentation policy in advance of
the actual acquisition process, we enhance the informativeness of the
generated virtual data instances. We believe that the proposed LADA framework
can improve the performance of deep learning models, especially when the
annotation by human experts is expensive.

## References

  * Ash et al. (2020) Ash, J. T.; Zhang, C.; Krishnamurthy, A.; Langford, J.; and Agarwal, A. 2020.  Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds.  In _ICLR_.
  * Chapelle et al. (2001) Chapelle, O.; Weston, J.; Bottou, L.; and Vapnik, V. 2001.  Vicinal risk minimization.  In _Advances in neural information processing systems_ , 416–422.
  * Cohn, Ghahramani, and Jordan (1996) Cohn, D. A.; Ghahramani, Z.; and Jordan, M. I. 1996.  Active learning with statistical models.  _Journal of artificial intelligence research_ 4: 129–145.
  * Cubuk et al. (2019) Cubuk, E. D.; Zoph, B.; Mane, D.; Vasudevan, V.; and Le, Q. V. 2019.  Autoaugment: Learning augmentation strategies from data.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 113–123.
  * Freeman (1965) Freeman, L. 1965.  _Elementary applied statistics: for students in behavioral science_.  Wiley.  URL https://books.google.co.kr/books?id=r4VRAAAAMAAJ.
  * Goodfellow et al. (2014) Goodfellow, I. J.; Pouget-Abadie, J.; Mirza, M.; Xu, B.; Warde-Farley, D.; Ozair, S.; Courville, A.; and Bengio, Y. 2014.  Generative Adversarial Nets.  In _Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 2_ , NIPS’14, 2672–2680. Cambridge, MA, USA: MIT Press.
  * Guo, Mao, and Zhang (2019) Guo, H.; Mao, Y.; and Zhang, R. 2019.  Mixup as locally linear out-of-manifold regularization.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 33, 3714–3722.
  * Hastings (1970) Hastings, W. K. 1970.  Monte Carlo sampling methods using Markov chains and their applications .
  * He et al. (2016) He, K.; Zhang, X.; Ren, S.; and Sun, J. 2016.  Deep residual learning for image recognition.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 770–778.
  * Houlsby et al. (2011) Houlsby, N.; Huszar, F.; Ghahramani, Z.; and Lengyel, M. 2011.  Bayesian Active Learning for Classification and Preference Learning.  _CoRR_ abs/1112.5745.
  * Jaderberg et al. (2015) Jaderberg, M.; Simonyan, K.; Zisserman, A.; et al. 2015.  Spatial transformer networks.  In _Advances in neural information processing systems_ , 2017–2025.
  * Jankowiak and Karaletsos (2019) Jankowiak, M.; and Karaletsos, T. 2019.  Pathwise Derivatives for Multivariate Distributions.  In Chaudhuri, K.; and Sugiyama, M., eds., _The 22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan_ , volume 89 of _Proceedings of Machine Learning Research_ , 333–342. PMLR.  URL http://proceedings.mlr.press/v89/jankowiak19a.html.
  * Jankowiak and Obermeyer (2018) Jankowiak, M.; and Obermeyer, F. 2018.  Pathwise Derivatives Beyond the Reparameterization Trick.  In Dy, J. G.; and Krause, A., eds., _Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018_ , volume 80 of _Proceedings of Machine Learning Research_ , 2240–2249. PMLR.  URL http://proceedings.mlr.press/v80/jankowiak18a.html.
  * Kingma and Welling (2014) Kingma, D. P.; and Welling, M. 2014.  Auto-Encoding Variational Bayes.  In Bengio, Y.; and LeCun, Y., eds., _2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings_.  URL http://arxiv.org/abs/1312.6114.
  * Krizhevsky, Hinton et al. (2009) Krizhevsky, A.; Hinton, G.; et al. 2009.  Learning multiple layers of features from tiny images .
  * Liu and Ferrari (2017) Liu, B.; and Ferrari, V. 2017.  Active learning for human pose estimation.  In _Proceedings of the IEEE International Conference on Computer Vision_ , 4363–4372.
  * Maaten and Hinton (2008) Maaten, L. v. d.; and Hinton, G. 2008.  Visualizing data using t-SNE.  _Journal of machine learning research_ 9(Nov): 2579–2605.
  * Netzer et al. (2011) Netzer, Y.; Wang, T.; Coates, A.; Bissacco, A.; Wu, B.; and Ng, A. Y. 2011.  Reading digits in natural images with unsupervised feature learning .
  * Perez and Wang (2017) Perez, L.; and Wang, J. 2017.  The effectiveness of data augmentation in image classification using deep learning.  _arXiv preprint arXiv:1712.04621_ .
  * Sener and Savarese (2018) Sener, O.; and Savarese, S. 2018.  Active Learning for Convolutional Neural Networks: A Core-Set Approach.  In _International Conference on Learning Representations_.
  * Settles (2009) Settles, B. 2009.  Active learning literature survey.  Technical report, University of Wisconsin-Madison Department of Computer Sciences.
  * Shannon (1948) Shannon, C. E. 1948.  A mathematical theory of communication.  _Bell Syst. Tech. J._ 27(3): 379–423.
  * Sinha, Ebrahimi, and Darrell (2019) Sinha, S.; Ebrahimi, S.; and Darrell, T. 2019.  Variational adversarial active learning.  In _Proceedings of the IEEE International Conference on Computer Vision_ , 5972–5981.
  * Tong (2001) Tong, S. 2001.  _Active learning: theory and applications_ , volume 1.  Stanford University USA.
  * Tran et al. (2019) Tran, T.; Do, T.; Reid, I. D.; and Carneiro, G. 2019.  Bayesian Generative Active Deep Learning.  In Chaudhuri, K.; and Salakhutdinov, R., eds., _Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA_ , volume 97 of _Proceedings of Machine Learning Research_ , 6295–6304. PMLR.  URL http://proceedings.mlr.press/v97/tran19a.html.
  * Verma et al. (2018) Verma, V.; Lamb, A.; Beckham, C.; Najafi, A.; Courville, A.; Mitliagkas, I.; and Bengio, Y. 2018.  Manifold mixup: Learning better representations by interpolating hidden states .
  * Xiao, Rasul, and Vollgraf (2017) Xiao, H.; Rasul, K.; and Vollgraf, R. 2017.  Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms.  _arXiv preprint arXiv:1708.07747_ .
  * Zhang et al. (2017) Zhang, H.; Cisse, M.; Dauphin, Y. N.; and Lopez-Paz, D. 2017.  mixup: Beyond empirical risk minimization.  _arXiv preprint arXiv:1710.09412_ .