# Quantitative Thomas-Yau uniqueness

Yang Li

###### Abstract

Under Floer theoretic conditions, we obtain quantitative estimates on the
closeness (Hausdorff distance, flat norm, F-metric) between two Lagrangians,
depending on the smallness of Lagrangian angles. Some applications include a
strong-weak uniqueness theorem for special Lagrangians, and a characterization
of varifold convergence to special Lagrangians in terms of Lagrangian angles.

##  1 Background and introduction

Inside an $n$-dimensional compact Kähler manifold $(X,\omega)$ with a nowhere
vanishing holomorphic volume form $\Omega$ such that
$\frac{\omega^{n}}{n!}=e^{2\rho}\frac{i^{n^{2}}}{2^{n}}\Omega\wedge\overline{\Omega}$,
an $n$-dimensional compact oriented submanifold $L$ is called special
Lagrangian, if

$\omega|_{L}=0,\quad\text{Im}\Omega|_{L}=0.$ (1)

These sit at the crossroad of minimal surface theory and symplectic geometry:

  * •

If the metric is Calabi-Yau (namely $\rho=0$), then special Lagrangians are
absolute volume minimizers inside their homology classes.

  * •

Lagrangian submanifolds equipped with (unobstructed) brane structures define
objects inside the Fukaya category.

###  1.1 Thomas-Yau uniqueness

As part of a wider program to relate the existence and uniqueness questions of
special Lagrangian branes, to Fukaya categories and stability condition,
Thomas and Yau [13] proved a remarkable uniqueness theorem, which from the
modern perspective reads

######  Theorem 1.1.

[13][7][8] Let $L,L^{\prime}$ be two compact embedded special Lagrangians with
unobstructed brane structures, defining isomorphic objects in the derived
Fukaya category, then their supports coincide.

Recall the Lagrangian angle $\theta:L\to\mathbb{R}$ is defined by
$\Omega|_{L}=e^{-\rho}e^{i\theta}dvol_{L}$, so special Lagrangians amount to
the condition $\theta=0$. The Floer degree of a transverse intersection point
$p\in CF^{*}(L_{1},L_{2})$ between two Lagrangians $L_{1},L_{2}$ is

$\mu_{L_{1},L_{2}}(p)=\frac{1}{\pi}(\sum_{1}^{n}\alpha_{i}-\theta_{L_{2}}(p)+\theta_{L_{1}}(p))\in\mathbb{Z},$
(2)

where inside $T_{p}X\simeq\mathbb{C}^{n}$ we can put the tangent spaces into
the standard form

$T_{p}L_{1}=\mathbb{R}^{n},\quad T_{p}L_{2}=(e^{i\alpha_{1}},\ldots
e^{i\alpha_{n}})\mathbb{R}^{n},\quad 0<\alpha_{i}<\pi.$

The proof idea of Theorem 1.1 can be outlined as follows. Assume $L\neq
L^{\prime}$.

  * •

By making $C^{\infty}$-small Hamiltonian perturbations of $L,L^{\prime}$, we
can replace $L,L^{\prime}$ by isomorphic objects $L_{1},L_{2}$ _intersecting
transversely_. Morever, either by a Morse theoretic argument [13], or using
real analyticity via a Lojasiewicz inequality [7], one can remove the degree
zero intersection points, to ensure the Floer cochain group
$CF^{0}(L_{1},L_{2})=0$.

  * •

Consequently the Floer cohomology $HF^{0}(L,L)\simeq HF^{0}(L_{1},L_{2})=0$.
In particular the unit of the Floer cohomology ring vanishes, which is
impossible, because it violates Poincaré duality (alternatively, because the
image of the unit under the open-closed map is the homology class $[L]\in
H_{n}(X,\Lambda_{nov})$, which cannot be zero).

We take note of a few conceptual features:

  * •

The proof of Thomas-Yau uniqueness is analogous to a _strong maximum
principle_ argument. The role of symplectic topology is to force the existence
of a degree zero intersection point, which is analogous to the existence of a
maximum, and a local calculation concerning the intersection point results in
a contradiction.

  * •

The metric is not necessarily Calabi-Yau. The compactness of $X$ can be
replaced by any other standard settings where Floer theory makes sense.

  * •

The proof is not sensitive to the details of the Fukaya category.

  * •

The known proofs rely essentially on smoothness assumptions of the
Lagrangians.

###  1.2 Quantitative Thomas-Yau uniqueness

A common theme in geometric analysis is _rigidity theorems_ , e.g. a strong
maximum principle naturally suggests a Harnack inequality. Analogously, we ask

######  Question 1.

Suppose $L,L^{\prime}$ have _small_ Lagrangian angles (eg.
$\left\lVert\theta\right\rVert_{C^{0}}\ll 1$, or
$\left\lVert\theta\right\rVert_{L^{1}}\ll 1$), then do they have to be
_uniformly_ close to each other (eg. in the Hausdorff distance for subsets, or
in the flat norm distance for integral currents, or the $F$-metric for
varifolds)?

######  Remark 1.

The very recent paper of Abouzaid and Imagi [1] studies symplectic topological
consequences of special Lagrangians lying inside a small _$C^{0}$
-neighbourhood_ of a given special Lagrangian, under additional hypotheses on
the fundamental group and its representations.

We now indicate why new ideas are needed for this question. In a naïve
strategy, one takes a sequence of $L_{i},L_{i}^{\prime}$, with Lagrangian
angles converging to zero, and attempts to use compactness theorems in
geometric measure theory to extract subsequential limits
$L_{\infty},L^{\prime}_{\infty}$, which should be special Lagrangian integral
currents. To deduce that $L_{i}$ is close to $L_{i}^{\prime}$, one would like
to prove $L_{\infty}=L^{\prime}_{\infty}$. This would require a _singular
Lagrangian_ version of the Thomas-Yau uniqueness theorem, which is yet
unknown. (The interested reader may see [6, section 5.7] for some heuristic
ideas).

Our main result is

######  Theorem 1.2.

Let $L,L^{\prime}$ be two compact, smoothly embedded Lagrangians with
unobstructed brane structures, in the same fixed homology classes in
$H_{n}(X,\mathbb{Q})$, such that $HF^{0}(L,L^{\prime})\neq 0$ or
$HF^{0}(L^{\prime},L)\neq 0$ holds. Assume there is some small enough
$\epsilon$, such that the Lagrangian angles satisfy

  * •

$\left\lVert\theta_{L}\right\rVert_{C^{0}}\leq\epsilon\ll 1.$

  * •

$\left\lVert\theta_{L^{\prime}}\right\rVert_{C^{0}}\leq\frac{\pi}{2}-\epsilon_{0}$
for some fixed $\epsilon_{0}>0$, and
$\text{Vol}(\\{\theta_{L^{\prime}}>\epsilon\\})\ll\epsilon^{n}$.

Then the Hausdorff distance between $L,L^{\prime}$ is uniformly small:

$\begin{cases}\sup_{p\in L}\text{dist}(p,L^{\prime})\leq
C\epsilon^{\frac{1}{2n}},\\\ \sup_{p\in L^{\prime}}\text{dist}(p,L)\leq
C\epsilon^{\frac{1}{4n^{2}}}.\end{cases}$ (3)

The flat norm distance between $L$ and $L^{\prime}$ is bounded uniformly by
$C\epsilon^{\frac{1}{4n}+\frac{1}{4n^{2}}}$. The $F$-metric between
$L,L^{\prime}$ is bounded by $C\epsilon^{\frac{1}{8n}}$. The constants depend
on $\epsilon_{0}$, $X$, and the $C^{2}$-regularity bounds on $L$, but not on
the regularity bounds on $L^{\prime}$, nor on the small $\epsilon$.

A few explanations may clarify the significance of the assumptions:

  * •

If $L,L^{\prime}$ are isomorphic nonzero objects in the derived Fukaya
category, then $HF^{0}(L,L^{\prime})\neq 0$, $HF^{0}(L^{\prime},L)\neq 0$ and
$[L]=[L^{\prime}]\in H_{n}(X,\mathbb{Q})$.

  * •

The condition
$\left\lVert\theta_{L^{\prime}}\right\rVert_{C^{0}}\leq\frac{\pi}{2}-\epsilon_{0}$
is called quantitative almost calibratedness, important in the Thomas-Yau
program (_cf._ [13][6]). Obviously, this is weaker than assuming the
$C^{0}$-norm on $\theta_{L^{\prime}}$ to be small. One of its immediate
consequences is the a priori homological mass bound

$\text{Vol}(L^{\prime})=\int_{L^{\prime}}dvol\leq\frac{1}{\sin\epsilon_{0}}\int_{L^{\prime}}\text{Re}\Omega.$

  * •

The weak $L^{1}$-type bound
$\text{Vol}(\\{\theta_{L^{\prime}}>\epsilon\\})\ll\epsilon^{n}$ is implied by
$\left\lVert\theta_{L^{\prime}}\right\rVert_{L^{1}(L^{\prime})}\ll\epsilon^{n+1}$.
This is again weaker than the $C^{0}$-smallness of $\theta_{L^{\prime}}$. This
weakening is attractive because if a sequence $L_{i}^{\prime}$ of
quantitatively almost calibrated Lagrangians converge to a special Lagrangian
in the weak topology of varifolds (_i.e._ as Radon measures on the
Grassmannian bundle), then
$\left\lVert\theta_{L^{\prime}_{i}}\right\rVert_{L^{1}}\to 0$ as $i\to\infty$,
but $\left\lVert\theta_{L^{\prime}_{i}}\right\rVert_{C^{0}}$ does not
necessarily converge to zero.

Theorem 1.2 can be interpreted as the quantitative version of a strong-weak
uniqueness theorem.

######  Definition 1.3.

Given a sequence of smooth embedded unobstructed Lagrangian branes $L_{i}$
which lie in a fixed nonzero derived Fukaya category class (so in particular
the same rational homology class), and are all quantitatively almost
calibrated. We say a Lagrangian $C^{2}$-submanifold (resp. Lagrangian integral
current) $L_{\infty}$ is a $C^{2}$-limit (resp. varifold limit) if $L_{i}$
converge to $L_{\infty}$ in the $C^{2}$-topology (resp. in both the weak
topology of varifolds and the flat norm topology of integral currents). Notice
there is no requirement that $L_{\infty}$ is equipped with any brane
structure.

######  Corollary 1.4.

(Strong-weak uniqueness) Fix a derived Fukaya category class, and let
$L_{\infty}$ be a $C^{2}$-limit and $L_{\infty}^{\prime}$ a varifold limit.
Assume both are special Lagrangian
$\theta_{L_{\infty}}=\theta_{L^{\prime}_{\infty}}=0$, then
$L_{\infty}=L_{\infty}^{\prime}$ as integral currents.

###### Proof.

From the $C^{2}$-limit assumption, $L_{i}\to L_{\infty}$ with
$\left\lVert\theta\right\rVert_{C^{0}}\to 0$. From the varifold limit
assumption, $L_{i}^{\prime}\to L$ with
$\left\lVert\theta\right\rVert_{L^{1}}\to 0$. All Lagrangians lie in the same
homology class, and all the approximants $L_{i},L_{i}^{\prime}$ lie in the
same derived category class. Thus Theorem 1.2 applies to
$L_{i},L_{i}^{\prime}$ for $i\gg 1$ and arbitrarily small $\epsilon$, and the
uniform $C^{2}$-regularity on $L_{i}$ means that all the constants are
uniform. The flat norm distance between $L_{i},L_{i}^{\prime}$ is bounded by
$C\epsilon^{\frac{1}{4n}+\frac{1}{4n^{2}}}$. Taking the limit as
$i\to+\infty$, the flat norm distance between $L_{\infty}$ and
$L_{\infty}^{\prime}$ is bounded by
$C\epsilon^{\frac{1}{4n}+\frac{1}{4n^{2}}}$, and letting $\epsilon\to 0$ gives
$L_{\infty}=L_{\infty}^{\prime}$. ∎

Provided a derived Fukaya category class admits a special Lagrangian
$C^{2}$-limit, we have a satisfactory characterization of varifold topology
convergence in terms of the Lagrangian angle function.

######  Corollary 1.5.

Fix a derived Fukaya category class, and assume there is a sequence of smooth
embedded representatives $L_{i}$, converging in $C^{2}$ topology to a special
Lagrangian $L_{\infty}$. Let $L_{i}^{\prime}$ be a sequence of quantitatively
almost calibrated Lagrangian branes in the same derived Fukaya category class.
Then $L_{i}^{\prime}$ converges in the varifold topology to $L_{\infty}$ if
and only if $\left\lVert\theta_{L_{i}^{\prime}}\right\rVert_{L^{1}}\to 0$.

###### Proof.

Since the Lagrangian angle is a continuous function on the open subset
$\\{\text{Re}\Omega>0\\}$ in the Grassmannian bundle, the weak topology
convergence of $L_{i}$ implies

$\int_{L_{i}}|\theta|dvol_{L_{i}}\to\int_{L_{\infty}}|\theta|dvol=0.$

Conversely, if $\left\lVert\theta_{L_{i}^{\prime}}\right\rVert_{L^{1}}\to 0$,
then Theorem 1.2 applies to $L_{i}$ and $L_{i}^{\prime}$ for $i\gg 1$ and
arbitrarily small $\epsilon$, whence the $F$-metric between $L_{i}$ and
$L_{i}^{\prime}$ is bounded by $C\epsilon^{1/8n}$ for $i\gg 1$ depending on
$\epsilon$. In the $i\to+\infty$ limit, the $F$-metric distance between
$L_{i}^{\prime}$ and $L_{\infty}$ tends to zero. Since all Lagrangians have
uniform mass bounds, the $F$-metric induces the same topology as the weak
topology on varifolds, whence $L_{i}^{\prime}\to L_{\infty}$ in the varifold
topology. ∎

######  Remark 2.

Since in our setting $\left\lVert\theta\right\rVert_{L^{\infty}}$ and the mass
of the Lagrangian both have uniform bounds, by interpolation
$\left\lVert\theta\right\rVert_{L^{1}}\to 0$ is equivalent to
$\left\lVert\theta\right\rVert_{L^{p}}\to 0$ for any fixed $p>0$.

###  1.3 Open questions

We raise some natural questions:

######  Question 2.

With the same assumptions on $L,L^{\prime}$ as in Thm 1.2, what would be the
_sharp exponent_ for the Hausdorff distance bound in terms of $\epsilon$?

This question has many variants. For instance, one can replace
$\text{Vol}(\\{\theta_{L^{\prime}}>\epsilon\\})\leq\epsilon^{n}$ with the
smallness of $\left\lVert\theta_{L^{\prime}}\right\rVert_{L^{p}(L^{\prime})}$
(resp. $\left\lVert\theta_{L^{\prime}}\right\rVert_{C^{0}(L^{\prime})}$). We
expect the exponents of $\epsilon$ in Theorem 1.2 to be not optimal. On the
other hand, even in the simple setting of Calabi-Yau metrics, assuming $L$ is
a fixed smooth special Lagrangian, and
$\left\lVert\theta_{L^{\prime}}\right\rVert_{L^{1}}<\epsilon$, we do not
expect that the Hausdorff distance to be uniformly bounded by $O(\epsilon)$.
The reason is that the linearization of the special Lagrangian equation is the
Poisson equation $\Delta u=f$, and the failure of the Sobolev embedding
$W^{2,1}\to C^{1}$ means we cannot expect $u$ to have $C^{1}$-bounds
proportional to $\left\lVert f\right\rVert_{L^{1}}$. Some nonlinear effects
are necessary for a result like Thm. 1.2.

######  Question 3.

Do the uniform constants really depend on the $C^{2}$-regularity bounds of
$L$?

The part of our arguments most sensitive to $C^{2}$-regularity bounds is Lemma
2.1, which involves solving the Poisson equation on $L$ with bounds. If $L$
has some mild degeneration, eg. when it is approximately the desingularization
of some special Lagrangian with certain isolated conical singularities, then
we expect results similar to Theorem 1.2 would hold, with possibly different
exponents for $\epsilon$. On the other hand, it is not clear if we can drop
all regularity bounds on $L$ beyond the information of the homology class and
the Lagrangian angles. A closely related question is whether _strong
uniqueness_ holds:

######  Question 4.

Consider the variant of Cor. 1.4 where both $L_{\infty}$ and
$L_{\infty}^{\prime}$ are varifold limits. Assume both are special Lagrangian
$\theta_{L_{\infty}}=\theta_{L^{\prime}_{\infty}}=0$, then do they coincide as
integral currents?

As we will indicate in a series of remarks, the embedded assumption on the
Lagrangians in Theorem 1.2 can be relaxed to immersed Lagrangians with
_connected domains_ and transverse self intersection points, with rather minor
changes. The connected domain assumption is quite crucial to our arguments
here (_cf._ Remark 3), and also appears in the literature on Thomas-Yau
uniqueness for immersed Lagrangians [8]. On the other hand, the hypothesis of
Theorem 1.2 is a little weaker than assuming $L,L^{\prime}$ to lie in the same
derived Fukaya category class, and does not make explicit use of holomorphic
curves.

######  Question 5.

If $L,L^{\prime}$ are immersed Lagrangians defining the same object in the
derived Fukaya category, not necessarily with connected domains, then is there
still an analogue of the quantitative Thomas-Yau uniqueness theorem? Is there
a corresponding strong-weak uniqueness theorem?

##  2 Proof of quantitative Thomas-Yau uniqueness

Throughout this section, let $L,L^{\prime}$ be two compact, smoothly embedded
Lagrangians with unobstructed brane structures, such that
$HF^{0}(L,L^{\prime})\neq 0$ or $HF^{0}(L^{\prime},L)\neq 0$. All constants
$C$ depend only on $X$ and the $C^{2}$-regularity bounds of $L$, but not on
the regularity bounds for $L^{\prime}$.

###  2.1 Small $\left\lVert\theta\right\rVert_{C^{0}}$ case

In the special case where $\left\lVert\theta\right\rVert_{C^{0}}\ll 1$ is
small, we show that the set of points on $L$ lying near $L^{\prime}$ is
sufficiently dense. This argument contains the Floer theoretic ingredient in
our strategy. We do not yet impose $[L]=[L^{\prime}]\in H_{n}(X)$.

######  Lemma 2.1.

Assume $\left\lVert\theta\right\rVert_{C^{0}}\leq\epsilon\ll 1$ for both
$L,L^{\prime}$. For all $p\in L$, there is some $p^{\prime}\in L\cap
B(C_{1}\epsilon^{1/2n})$, such that $\text{dist}(p^{\prime},L^{\prime})\leq
C\epsilon^{\frac{n+1}{2n}}$. In particular $\text{dist}(p,L^{\prime})\leq
C\epsilon^{1/2n}$.

###### Proof.

We perform a Hamiltonian perturbation of the $L$ within the Weinstein tubular
neighbourhood $T^{*}L$ of the Lagrangian $L$. For a smooth Hamiltonian
function $h:X\to\mathbb{R}$, the time $\epsilon$ flow sends $L$ to
$\varphi_{\epsilon}(L)$ whose Lagrangian angle
$\theta_{\varphi_{\epsilon}(L)}$ satisfies

$|\theta_{\varphi_{\epsilon}(L)}-\theta_{L}-\epsilon\mathcal{L}(h)|\leq
C\epsilon^{2}(\left\lVert dh\right\rVert_{C^{1}}^{2}+\left\lVert
dh\right\rVert_{C^{1}}),$

where the linear operator $\mathcal{L}$ is the weighted Laplacian prescribed
by

$\mathcal{L}(h)e^{-\rho}dvol=d(e^{-\rho}*_{L}dh),$

which in the Calabi-Yau metric case ($\rho=0$) reduces to the Laplacian.

Given any point $p\in L$, we prescribe a smooth function $f:L\to\mathbb{R}$,
such that $f=4$ outside $B(p,\eta)$ for some $\eta\ll 1$ to be determined, the
weighted integral $\int_{L}e^{-\rho}fdvol_{L}=0$, and $|f|+|df|\eta\leq
C\eta^{-n}$ inside $B(p,\eta)$. We solve $\mathcal{L}(h)=f$ with
$\int_{L}he^{-\rho}dvol_{L}=0$, to find the smooth function $h$ with bound
$|dh|+\eta|\nabla dh|\leq C\eta^{1-n}.$ Using the smoothness of $L$, we can
extend $h$ to a smooth function on $X$ with the same bounds.

We choose $\eta=C_{1}\epsilon^{1/2n}$ for some large constant $C_{1}$
independent of $\epsilon$. Then the dispacement of the time $\epsilon$-flow is
bounded by $C\epsilon|dh|\leq C\epsilon\eta^{1-n}\leq
C\epsilon^{\frac{n+1}{2n}}\ll 1$, so $\varphi_{\epsilon}(L)$ is well defined
inside the Weinstein neighbourhood. Furthermore,

$|\theta_{\varphi_{\epsilon}(L)}-\theta_{L}-\epsilon\mathcal{L}(h)|\leq
C\epsilon^{2}(\left\lVert dh\right\rVert_{C^{1}}^{2}+\left\lVert
dh\right\rVert_{C^{1}})\leq C(\epsilon\eta^{-n})^{2}\ll\epsilon,$

whence on $\varphi_{\epsilon}(L\setminus B(p,\eta))$,

$\theta_{\varphi_{\epsilon}(L)}\geq\theta_{L}+\epsilon\mathcal{L}(h)-\epsilon\geq\epsilon\mathcal{L}(h)-2\epsilon\geq
4\epsilon-2\epsilon\geq 2\epsilon.$

Without loss $\varphi_{\epsilon}(L)$ is transverse to $L^{\prime}$ by
genericity. By formula (2), any Lagrangian intersection $q\in
CF^{*}(\varphi_{\epsilon}(L),L^{\prime})$ has Floer degree

$\mu(q)\geq\frac{1}{\pi}(\theta_{\varphi_{\epsilon}(L)}-\theta_{L^{\prime}})\geq\frac{1}{\pi}\epsilon,$

hence degree zero intersections cannot occur on $\varphi_{\epsilon}(L\setminus
B(p,\eta))$. By the Hamiltonian invariance of Floer cohomology

$HF^{0}(\varphi_{\epsilon}(L),L^{\prime})=HF^{0}(L,L^{\prime})\neq 0.$

This forces there to be an intersection $q\in\varphi_{\epsilon}(L\cap
B(p,\eta))\cap L^{\prime}$. Thus there is $p^{\prime}\in L\cap B(p,\eta)$
whose distance to $q$ is less than $C\epsilon|dh|\leq
C\epsilon^{\frac{n+1}{2n}}$. ∎

######  Remark 3.

If we replace the embedded Lagrangians by immersed Lagrangians with connected
domain and transverse self intersections, we can still use the Floer
cohomology of Akaho-Joyce [2]. The main caveat is that in order for the
Hamiltonian function $h:L\to\mathbb{R}$ to extend to $X$, we need $h$ to take
the same value at the finitely many self intersection points. These finitely
many linear constraints are easy to meet, by relaxing $f=4$ outside
$B(p,\eta)$ to the more flexible condition $f\geq 4$ outside $B(p,\eta)$.

The connected domain hypothesis on $L$ is important for solving the equation
$\mathcal{L}(h)=f$. If we drop this hypothesis, then take for instance $L$ the
disjoint union of two special Lagrangians, and $L^{\prime}$ to be one of the
components, and the lemma would be false.

######  Remark 4.

Lemma 2.1 easily gives a _new proof_ to the original Thomas-Yau uniqueness
theorem 1.1, not relying on Morse theory or real analyticity. Indeed, by
taking the $\epsilon\to 0$ limit, we deduce that each point on $L$ has zero
distance to $L^{\prime}$, hence $L\subset L^{\prime}$. Since the roles of
$L,L^{\prime}$ are symmetric in theorem 1.1, we recover $L=L^{\prime}$. The
rest of this paper treats the additional difficulties caused by giving up
quantitative regularity bounds on $L^{\prime}$, and by relaxing the
$C^{0}$-smallness of $\theta_{L^{\prime}}$ to merely smallness on most of the
measure.

###  2.2 Generic perturbation technique

The technique of controlling the number of intersection points by utilizing a
sufficiently rich family of perturbations, originated from Arnold [4], who
used it to study the dynamical growth of intersection points. Prop. 2.2 is a
detailed exposition to clarify the dependence of constants. Seidel [11,
Lecture 5] contains an application to bound the rank of Lagrangian Floer
cohomology.

Let $M$ be an $m$-dimensional compact manifold, and $N,N^{\prime}$ be two
compact submanifolds of complementary dimension, and let $U\subset N^{\prime}$
be an open subset. Let $V$ be a $p$-dimensional space of vector fields on $M$,
which is surjective to the normal space $TM/TN$ at every point of $N$. Fix a
Euclidean metric on $V$, and consider the $p$-dimensional family of
diffeomorphism $\phi_{t}$ of $M$ obtained by exponentiating the vector fields
in $P=B(0,\epsilon)\subset V$ for some small $\epsilon$. This induces a
$p$-parameter deformation family $\mathcal{N}\to P$ for $N$, whose fibres are
$\phi_{t}(N)$ for $t\in P$. There is an evaluation map $\pi:\mathcal{N}\to M$,
which by construction is surjective on tangent spaces.

######  Proposition 2.2.

There exists some $t\in P$, such that $\phi_{t}(N)$ intersects $N^{\prime}$
transversely, and the number of intersection points

$|\phi_{t}(N)\cap U|\leq C\epsilon^{-n}\text{Vol}(U),$

where the constant $C$ depends only on $M$, $V$, and the $C^{1}$-regularity of
$N$, but not on the small $\epsilon$ and the regularity bounds of
$N^{\prime}$.

###### Proof.

The transversality holds for a.e. $t\in P$ by Sard-Smale, so that
$|\phi_{t}(N)\cap U|$ is a well defined integer for a.e. $t\in P$. The
integral

$\int_{P}|\phi_{t}(N)\cap U|dt\leq\text{Vol}(\mathcal{N}\cap\pi^{-1}(U)).$

Now $\pi:\mathcal{N}\cap\pi^{-1}(U)\to U$ is surjective on tangent spaces,
with Jacobian factor bounded below by $C^{-1}$, so

$\text{Vol}(\mathcal{N}\cap\pi^{-1}(U))\leq
C\int_{U}\text{Vol}(\mathcal{N}\cap\pi^{-1}(y))dy.$

For each $y\in U$, the contributions to $\mathcal{N}\cap\pi^{-1}(y)$ come from
the deformations of the small local region $N\cap B(y,C\epsilon)$. The
assumption that $V$ is surjective to $TM/TN$ at every point of $N$, together
with the $C^{1}$-regularity of $N$, allows us to find a codimension $n$
subspace $V_{y}\subset V$, such that the projection

$\mathcal{N}\cap\pi^{-1}(y)\to P=B(0,\epsilon)\subset V\to V_{y}$

exhibits $\mathcal{N}\cap\pi^{-1}(y)$ as part of a $C^{1}$-graph over the
$\epsilon$-ball inside $V_{y}$, whose volume is bounded by $C\epsilon^{p-n}$.
We deduce

$\text{Vol}(\mathcal{N}\cap\pi^{-1}(y))\leq C\epsilon^{p-n}.$

Combining the above,

$\int_{P}|\phi_{t}(N)\cap U|dt\leq C\epsilon^{p-n}\text{Vol}(U),$

so we can select some $t\in P$ with

$|\phi_{t}(N)\cap U|\leq C\text{Vol}(U)\epsilon^{p-n}(\text{Vol}(P))^{-1}\leq
C\text{Vol}(U)\epsilon^{-n}.$

∎

In our application, the idea is that under sufficiently generic Hamiltonian
deformation, subsets with very small measure do not contribute to Lagrangian
intersection points. This allows us to relax Lemma 2.1 to a weak
$L^{1}$-assumption on the Lagrangian angle of $L^{\prime}$.

######  Proposition 2.3.

Assume $\left\lVert\theta_{L}\right\rVert_{C^{0}}\leq\epsilon$, while
$\text{Vol}(\\{|\theta_{L^{\prime}}>\epsilon\\})\ll\epsilon^{-n}$. For all
$p\in L$, there is some $p^{\prime}\in L\cap B(C_{1}\epsilon^{1/2n})$, such
that $\text{dist}(p^{\prime},L^{\prime})\leq C\epsilon^{\frac{n+1}{2n}}$.

###### Proof.

We can find a large dimensional vector space $V$ of Hamiltonian vector fields,
which is surjective to the normal space $TX/TL$ at every point of $L$. We can
pick the Euclidean metric on $V$, so that for any $v$ in the unit ball of $V$,
the induced vector field acting on the Grassmannian bundle over $X$ has
$C^{0}$-norm $\ll 1$. In particular, the Hamiltonian diffeomorphisms
$\phi_{t}$ obtained via exponentiating vector fields in
$P=B(0,\epsilon)\subset V$, only change the Lagrangian angle function by an
amount $\ll\epsilon$.

We now re-examine the argument of Lemma 2.1. For any $p\in L$, we can
construct the $C^{1}$-small Hamiltonian deformation $\varphi_{\epsilon}(L)$ of
$L$, such that on $\varphi_{\epsilon}(L)(L\setminus B(p,\eta))$, we have
$\theta_{\varphi_{\epsilon}(L)}\geq 2\epsilon$. Thus for any $t\in P$, the
Hamiltonian deformation $\phi_{t}\varphi_{\epsilon}(L)$ satisfies away from
$B(p,\eta)$

$\theta_{\phi_{t}\varphi_{\epsilon}(L)}\geq\theta_{\varphi_{\epsilon}(L)}-\frac{\epsilon}{2}\geq\frac{3}{2}\epsilon.$

Let $U=\\{\theta_{L^{\prime}}>\epsilon\\}\subset L^{\prime}$, then according
to Prop. 2.2, there is some $t\in P$, such that
$\phi_{t}\varphi_{\epsilon}(L)$ is transverse to $L^{\prime}$, and the number
of intersection points

$|\phi_{t}\varphi_{\epsilon}(L)\cap U|\leq C\epsilon^{-n}\text{Vol}(U)\ll 1.$

Thus $\phi_{t}\varphi_{\epsilon}(L)\cap U$ is in fact _empty_. This forces
$\theta_{L^{\prime}}\leq\epsilon$ at the Lagrangian intersections
$\phi_{t}\varphi_{\epsilon}(L)\cap L^{\prime}$, and we conclude as in Lemma
2.1. ∎

Consequently, we get one half of the Hausdorff distance estimate:

######  Corollary 2.4.

Under the same conditions,

$\sup_{p\in L}\text{dist}(p,L^{\prime})\leq C\epsilon^{1/2n}.$ (4)

######  Remark 5.

We have so far not used the condition $[L]=[L^{\prime}]\in
H_{n}(X,\mathbb{Q})$. Without this condition, the constant $C$ in Cor. 2.4
really requires some regularity bound on $L$. For instance, we consider a
surgery exact triangle $L_{1}\to L_{2}\to L_{3}\to L_{1}[1]$, where $L_{2}$ is
the Lagrangian connected sum of $L_{1},L_{3}$ with very small neck, and all
Lagrangians angles can be made arbitrarily small in $C^{0}$. The mere
assumption that $HF^{0}(L_{1},L_{2})\neq 0$ cannot imply that $\sup_{p\in
L_{2}}\text{dist}(p,L_{1})$ is small. The proof breaks down because the
$C^{1}$-regularity bound of $L_{2}$ is highly degenerate in the neck region.

###  2.3 Monotonicity inequality

In minimal surface theory, the famous monotonicity formula says that for an
$n$-dimensional submanifold $N$ inside a smooth compact ambient manifold, if
the mean curvature $\left\lVert\vec{H}\right\rVert_{L^{\infty}}<+\infty$, then
inside coordinate balls the volume ratio

$e^{Cr}r^{-n}\text{Vol}(B(p,r))$

increases with the radius $r$. One useful consequence is that the volume of
$N\cap B(p,r)$ has a lower bound for $p\in N$. In this paper we _only impose
assumptions on the Lagrangian angle $\theta$, but not on $|\vec{H}|$
directly_, but the volume lower bound is still satisfied. Our style of
arguments are inspired by Neves [10, section 3.3] (_cf._ also [6, section
5.2]).

Recall a special case of the optimal isoperimetric inequality of Almgren.
Denote $\omega_{n}$ as the volume of the $n$-dim unit ball in
$\mathbb{R}^{n}$.

######  Proposition 2.5.

[3, Thm 10] Let $T$ be an $(n-1)$-dimensional integral current inside
$\mathbb{R}^{N}$ with $\partial T=0$. Then there is an integral $n$-current
$Q$ inside $\mathbb{R}^{N}$ with $\partial Q=T$ and

$Mass(Q)\leq
n^{-\frac{n}{n-1}}\omega_{n}^{-\frac{1}{n-1}}Mass(T)^{\frac{n}{n-1}}.$

######  Remark 6.

The inequality is saturated by the $n$-dimensional unit ball; this sharpness
of constant will be important. Morever, if $T$ is contained in some ball
$B(R)$, then $Q$ can also be taken inside $B(R)$, because there is a
retraction of $\mathbb{R}^{N}$ to $B(R)$ with Lipschitz constant one.

We now work inside the Kähler manifold $X$. Around any fixed $p\in X$, we can
take local holomorphic coordinates $z_{1},\ldots z_{n}$, such that for small
$|z|$,

$\omega=\frac{\sqrt{-1}}{2}\sum dz_{i}\wedge d\bar{z}_{i}+O(|z|),\quad
e^{\rho(p)}\Omega=dz_{1}\wedge\ldots dz_{n}+O(|z|),\quad
g=\sum|dz_{i}|^{2}+O(|z|).$

Beware that the smooth function $\rho$ measures the failure of the metric to
be Calabi-Yau.

######  Proposition 2.6.

Let $L^{\prime}$ be a smooth Lagrangian with
$|\theta|\leq\frac{\pi}{2}-\epsilon_{0}$, and there exists $p^{\prime}\in
L^{\prime}$ with $|p^{\prime}|\leq\lambda\ll 1$. We shall use
$C(\epsilon_{0})$ to denote constants depending only on $\epsilon_{0}$ and the
coordinate chart.

  1. 1.

For the Euclidean balls $B_{Eucl}(r)$ with radius $r>\lambda$ contained in the
chart, we have the volume lower bound $\text{Vol}(L^{\prime}\cap
B_{Eucl}(r))\geq C(\epsilon_{0})(r-\lambda)^{n}$.

  2. 2.

Assume furthermore that $\text{Vol}(\\{\theta_{L^{\prime}}>\epsilon\\}\cap
L^{\prime})\leq\lambda^{n}$ where $\lambda\geq\epsilon^{2}$. Then for the
Euclidean balls $B_{Eucl}(r)$ with radius $r>C(\epsilon_{0})\lambda$ contained
in the chart, we have the sharper volume lower bound

$\text{Vol}(L^{\prime}\cap
B_{Eucl}(r))\geq\omega_{n}(r-C(\epsilon_{0})\lambda)^{n}(1-C(\epsilon_{0})r).$
(5)

###### Proof.

For a.e. $0<r$ smaller than the radius of the coordinate ball (of order
$O(1)$), the level set $L^{\prime}\cap\partial B_{Eucl}(r)$ is smooth, and the
isoperimetric inequality allows us to find $Q\subset B_{Eucl}(r)$ with
$\partial Q=L^{\prime}\cap\partial B_{Eucl}(r)$ and mass bound

$Mass(Q)\leq
n^{-\frac{n}{n-1}}\omega_{n}^{\frac{-1}{n-1}}\mathcal{H}^{n-1}(L^{\prime}\cap
B_{Eucl}(r))^{\frac{n}{n-1}}.$

Since $\partial(L^{\prime}\cap B_{Eucl}(r))=\partial Q$, the form
$\text{Re}\Omega$ is closed, and
$\text{Re}\Omega|_{Q}\leq(1+O(r))e^{-\rho(p)}dvol_{Q}$,

$\int_{L^{\prime}\cap
B_{Eucl}(r)}\text{Re}\Omega=\int_{Q}\text{Re}\Omega\leq(1+O(r))e^{-\rho(p)}Mass(Q).$

Under the quantitative almost calibrated condition
$|\theta|\leq\frac{\pi}{2}-\epsilon_{0}$,

$\text{Vol}(L^{\prime}\cap B_{Eucl}(r))=\int_{L^{\prime}\cap
B_{Eucl}(r)}dvol\leq\frac{1}{\sin\epsilon_{0}}\int_{L^{\prime}\cap
B_{Eucl}(r)}e^{\rho}\text{Re}\Omega.$

Combining the above,

$\text{Vol}(L^{\prime}\cap B_{Eucl}(r))\leq
C(\epsilon_{0})\mathcal{H}^{n-1}(L^{\prime}\cap B_{Eucl}(r))^{\frac{n}{n-1}}.$

The function $f(r)=\text{Vol}(L^{\prime}\cap B_{Eucl}(r))$ is increasing in
$r$. Using the coarea formula, we rewrite the differential inequality as

$f(r)\leq C(\epsilon_{0})f^{\prime}(r)^{\frac{n}{n-1}}.$

For $r>\lambda$, we have $f(r)>0$, and the increasing function $f$ satisfies

$(f^{1/n})^{\prime}\geq C(\epsilon_{0}),$

whence $f\geq C(\epsilon_{0})(r-\lambda)^{n}$.

Next we assume $\text{Vol}(\\{\theta_{L^{\prime}}>\epsilon\\}\cap
L^{\prime})\leq\lambda^{n}$ and deduce the sharper volume lower bound. For
$O(1)>r>C\lambda$ with $C$ depending only on the volume lower bound constant,
we have $f(r)>\lambda^{n}$. Notice that on
$L^{\prime}\cap\\{\theta_{L^{\prime}}\leq\epsilon\\}\cap B_{Eucl}(r)$,

$\int_{L^{\prime}\cap
B_{Eucl}(r)\cap\\{\theta_{L^{\prime}}\leq\epsilon\\}}dvol\leq\frac{1}{\cos\epsilon}\int_{L^{\prime}\cap
B_{Eucl}(r)\cap\\{\theta_{L^{\prime}}\leq\epsilon\\}}e^{\rho}\text{Re}\Omega\leq\frac{1}{\cos\epsilon}\int_{L^{\prime}\cap
B_{Eucl}(r)}e^{\rho}\text{Re}\Omega.$

Whence

$\begin{split}&f(r)\leq\lambda^{n}+\frac{1}{\cos\epsilon}\int_{L^{\prime}\cap
B_{Eucl}(r)}e^{\rho}\text{Re}\Omega\\\
&\leq\lambda^{n}+\frac{1}{\cos\epsilon}(1+O(r))\text{Mass}(Q)\\\
&\leq\lambda^{n}+\frac{1}{\cos\epsilon}(1+O(r))n^{-\frac{n}{n-1}}\omega_{n}^{\frac{-1}{n-1}}f^{\prime}(r)^{\frac{n}{n-1}}.\end{split}$

Now $\frac{1}{\cos\epsilon}=1+O(\epsilon^{2})=1+O(\lambda)=1+O(r)$, so the
$\frac{1}{\cos\epsilon}$ factor can be absorbed into the $1+O(r)$ factor by
changing the constant. Rearranging the terms,

$f^{\prime}(r)\geq n\omega_{n}^{1/n}(1-O(r))(f-\lambda^{n})^{\frac{n-1}{n}},$

we deduce

$\frac{d}{dr}(f-\lambda^{n})^{\frac{1}{n}}\geq\omega_{n}^{1/n}(1-O(r)),$

and after integration

$f\geq\omega_{n}(r-C\lambda)^{n}(1-O(r))$

as required. ∎

######  Remark 7.

In the context of special Lagrangians in $\mathbb{C}^{n}$, a standard way to
deduce the monotonicity formula is to compare the volume of $L^{\prime}\cap
B(r)$ with the cone over $L^{\prime}\cap\partial B(r)$. If we follow this
strategy for smooth Lagrangians inside $\mathbb{C}^{n}$ with
$|\theta|\leq\epsilon$, then we would deduce

$\text{Vol}(L^{\prime}\cap B(r))\leq\frac{1}{\cos\epsilon}\int_{L^{\prime}\cap
B(r)}\text{Re}\Omega\leq\frac{r}{n\cos\epsilon}\mathcal{H}^{n-1}(L^{\prime}\cap
B(r)),$

whence $\frac{d}{dr}\log\text{Vol}(L^{\prime}\cap
B(r))\geq\frac{n\cos\epsilon}{r}$. Unfortunately, this would only prove the
monotonicity of $\text{Vol}(L^{\prime}\cap B(r))r^{-n\cos\epsilon}$, which is
_not sufficient_ to deduce the volume lower bound above.

We think it is an interesting question whether
$e^{Cr}\text{Vol}(L^{\prime}\cap B(r))r^{-n}$ is monotone under the mere
hypothesis that $\left\lVert\theta\right\rVert_{C^{0}}$ is small, with no
assumption directly on the mean curvature.

###  2.4 Hausdorff distance bound

We still need to show that all the points of $L^{\prime}$ must stay close to
$L$. The strategy is to first show that such points occupy almost the full
measure of $L^{\prime}$, and then use a monotonicity inequality argument to
extend this to all points.

We record a weighted volume upper bound:

######  Lemma 2.7.

If
$\text{Vol}(L^{\prime}\cap\\{\theta_{L^{\prime}}>\epsilon\\})\leq\epsilon^{n}$,
then
$\int_{L^{\prime}}e^{-\rho}dvol_{L^{\prime}}\leq\frac{1}{\cos\epsilon}\int_{L^{\prime}}\text{Re}\Omega+C\epsilon^{n}$.

###### Proof.

On the subset $L^{\prime}\cap\\{\theta_{L^{\prime}}\leq\epsilon\\}$, we
integrate
$e^{-\rho}dvol_{L^{\prime}}\leq\frac{1}{\cos\epsilon}\text{Re}\Omega|_{L^{\prime}}$.
∎

######  Lemma 2.8.

In the setting of Theorem 1.2,

$\int_{L^{\prime}\cap\\{\text{dist}(\cdot,L)\leq\epsilon^{1/4n}\\}}e^{-\rho}dvol_{L^{\prime}}\geq(1-C\epsilon^{1/{4n}})\int_{L}\text{Re}\Omega.$
(6)

###### Proof.

We can take $\lambda=C\epsilon^{1/2n}$, so that by Cor. 2.4, any $p\in L$ lies
within distance $\lambda$ to $L^{\prime}$. Our setting implies
$\lambda\geq\epsilon^{2}$ and
$\text{Vol}(\\{\theta_{L^{\prime}}>\epsilon\\})\leq\lambda^{n}$. By Prop. 2.6,
and the fact that Euclidean balls and Riemannian balls only differ by relative
error $O(r)$, we see that for $O(1)\geq r\geq C\lambda$,

$\text{Vol}_{g}(L^{\prime}\cap
B_{g}(p,r))\geq\omega_{n}(r-C\lambda)^{n}(1-Cr),\quad\forall p\in L.$

The constants are uniform for $p\in L$, so after integration

$\begin{split}&\int_{L}e^{-\rho(p)}\text{Vol}_{g}(L^{\prime}\cap
B_{g}(p,r))dvol_{L}(p)\\\
\geq&\omega_{n}(r-C\lambda)^{n}(1-Cr)\int_{L}e^{-\rho}dvol_{L}\\\
\geq&\omega_{n}(r-C\lambda)^{n}(1-Cr)\int_{L}\text{Re}\Omega.\end{split}$

The LHS can be computed by Fubini’s theorem as

$\int_{L^{\prime}}e^{-\rho(q)}dvol_{L^{\prime}}(q)\int_{L\cap
B_{g}(q,r)}e^{\rho(q)-\rho(p)}dvol_{L}(p).$

The function $\rho$ is smooth, so $|\rho(q)-\rho(p)|\leq Cr$. The Lagrangian
$L$ has fixed $C^{1}$-regularity bound, so for small radius $r\ll 1$,

$\int_{L\cap B_{g}(q,r)}dvol_{L}(p)\leq(1+Cr)\omega_{n}r^{n}.$

Combining the above,

$\begin{split}&\omega_{n}(r-C\lambda)^{n}(1-Cr)\int_{L}\text{Re}\Omega\leq\int_{L}e^{-\rho(p)}\text{Vol}_{g}(L^{\prime}\cap
B_{g}(p,r))dvol_{L}(p)\\\
\leq&(1+Cr)\omega_{n}r^{n}\int_{L^{\prime}\cap\\{\text{dist}(\cdot,L)\leq
r\\}}e^{-\rho(q)}dvol_{L^{\prime}}(q),\end{split}$

whence for $C\lambda\leq r\leq O(1)$,

$\int_{L^{\prime}\cap\\{\text{dist}(\cdot,L)\leq
r\\}}e^{-\rho(q)}dvol_{L^{\prime}}(q)\geq(1-Cr-C\frac{\lambda}{r})\int_{L}\text{Re}\Omega.$
(7)

The choice $r=\epsilon^{1/4n}$ yields the claim. ∎

######  Remark 8.

In the above, we assumed $L$ is embedded. If $L$ is merely immersed with
transverse self intersections, then the inequality

$\int_{L\cap B_{g}(q,r)}dvol_{L}(p)\leq(1+Cr)\omega_{n}r^{n}.$

will break down for $q$ in the $r$-neighbourhood of the self intersection
points. Essentially the same proof would then give

$\int_{L^{\prime}\cap\\{\text{dist}(\cdot,L)\leq\epsilon^{1/4n}\\}}e^{-\rho}dvol_{L^{\prime}}\geq(1-C\epsilon^{1/{4n}})\int_{L}\text{Re}\Omega-
Cr^{n}.$

Since $r^{n}\ll\epsilon^{1/4n}$, the $r^{n}$ term can be absorbed.

We can finally prove the remaining half of the Hausdorff distance bound:

######  Proposition 2.9.

In the setting of Theorem 1.2,

$\sup_{p\in L^{\prime}}\text{dist}(p,L)\leq C\epsilon^{\frac{1}{4n^{2}}}.$ (8)

###### Proof.

The assumption that $[L]=[L^{\prime}]\in H_{n}(X,\mathbb{Q})$, together with
Lemma 2.7 imply

$\int_{L^{\prime}}e^{-\rho}dvol_{L^{\prime}}\leq\frac{1}{\cos\epsilon}\int_{L}\text{Re}\Omega+C\epsilon^{n}\leq(1+C\epsilon^{2})\int_{L}\text{Re}\Omega.$

Combined with Lemma 6,

$\int_{L^{\prime}\cap\\{\text{dist}(\cdot,L)\geq\epsilon^{1/4n}\\}}e^{-\rho}dvol_{L^{\prime}}\leq
C\epsilon^{1/{4n}}\int_{L}\text{Re}\Omega.$

Assume $p\in L^{\prime}$ with $\text{dist}(p,L)=r_{p}>\epsilon^{1/4n}$, so
that $L^{\prime}\cap B(p,r_{p}-\epsilon^{1/4n})$ is contained inside
$L^{\prime}\cap\\{\text{dist}(\cdot,L)\geq\epsilon^{1/4n}\\}$. From Prop. 2.6,

$C(r_{p}-\epsilon^{1/4})^{n}\leq\text{Vol}(L^{\prime}\cap
B(p,r_{p}-\epsilon^{1/4n}))\leq C\epsilon^{1/{4n}}\int_{L}\text{Re}\Omega.$

Consequently $r_{p}\leq C\epsilon^{1/4n^{2}}$. ∎

###  2.5 Flat norm distance bound

Recall the flat norm distance between $L$ and $L^{\prime}$ is defined as

$\inf\\{\text{Mass}(T)+\text{Mass}(R):L-L^{\prime}=\partial T+R\\},$

where $T,R$ are integral currents of dim $n+1$ and $n$ respectively.
Intuitively, the Hausdorff distance bound is an $L^{\infty}$ type bound on the
distance function, while the flat norm behaves like an $L^{1}$ type bound. In
the Corollary below we leverage the previous information to get a slightly
better exponent than the obvious bound from $L^{\infty}\to L^{1}$, but we
expect this is still not sharp.

######  Corollary 2.10.

In the setting of Theorem 1.2, there is an integral current $T$ with $\partial
T=L^{\prime}-L$, such that $\text{Mass}(T)\leq
C\epsilon^{\frac{1}{4n^{2}}+\frac{1}{4n}}$.

###### Proof.

The Hausdorff bound shows that $L^{\prime}$ lies within a small
$C^{0}$-neighbourhood of $L$. Using the controlled $C^{1}$-regularity of $L$,
we can view the $C^{0}$-neighbourhood as its normal bundle, and obtain a
projection map $\pi:L^{\prime}\to L$. Since $L$ is homologous to $L^{\prime}$
by assumption, $\pi$ has degree one. We form an $(n+1)$-dimensional integral
current $T$ from the union of the line segments joining the points $q\in
L^{\prime}$ to $\pi(q)\in L$. Up to the choice of orientation sign, $\partial
T=L^{\prime}-\pi_{*}(L^{\prime})=L^{\prime}-L$. The lengths of the segments
are comparable to $\text{dist}(q,L)$. Thus

$\text{Mass}(T)\leq
C\int_{L^{\prime}}\text{dist}(\cdot,L)dvol_{L^{\prime}}=C\int_{0}^{\sup\text{dist}}dr\int_{L^{\prime}\cap\\{\text{dist}(\cdot,L)\geq
r\\}}dvol_{L^{\prime}}.$

Recall the weak $L^{1}$-type estimate (7), which implies

$\int_{L^{\prime}\cap\\{\text{dist}(\cdot,L)\geq r\\}}dvol_{L^{\prime}}\leq
C\begin{cases}1,\quad r=O(\epsilon^{1/2n}),\\\
\frac{\epsilon^{1/2n}}{r},\quad\epsilon^{1/2n}\leq r\leq\epsilon^{1/4n},\\\
\epsilon^{1/4n},\quad r>\epsilon^{1/4n}.\end{cases}$

By the Hausdorff distance estimate, $\sup\text{dist}(\cdot,L)\leq
C\epsilon^{1/4n^{2}}$. Combining the above gives the mass bound. ∎

######  Remark 9.

If $L$ is immersed rather than embedded, then $\pi$ would be only Lipschitz
near the self intersection points, but the above argument is unaffected.

###  2.6 F-metric bound

We view the $C^{0}$ neighbourhood of $L$ as its normal bundle, which has a
projection $\pi$ to its zero section $L$. Normal geodesic flow allows us to
identify the tangent spaces $T_{q}X$ with $T_{\pi(q)}X$. We now show that
$L^{\prime}$ is graphical over $L$ with small norm, away from a set with small
measure. Denote $\vec{T}(q)$ as the unit $n$-vector $e_{1}\wedge\ldots e_{n}$
on $L^{\prime}$ where $e_{i}$ is an oriented orthonormal frame of
$T_{q}L^{\prime}$, and $\vec{T}(\pi(q))$ as the analogous unit $n$-vector from
the oriented orthonormal basis of $T_{\pi(q)}L$.

######  Proposition 2.11.

In the setting of Theorem 1.2, away from a subset $E\subset L$ with
$\text{Vol}(E)+\text{Vol}(\pi^{-1}(E))\leq C\epsilon^{1/4n}$, the projection
$\pi:L^{\prime}\setminus\pi^{-1}(E)\to L\setminus E$ is bijective, and

$\int_{L^{\prime}\setminus\pi^{-1}(E)}|\vec{T}(q)-\vec{T}(\pi(q))|^{2}dvol(q)\leq
C\epsilon^{1/4n}.$

###### Proof.

The projection $\pi:L^{\prime}\to L$ is almost metric decreasing:

$|d\pi|_{T_{q}L^{\prime}}|\leq 1+O(\text{dist}(q,\pi(q))).$

As a general rule, the error coming from comparing ambient Kähler structures
at $q$ and $\pi(q)$ is bounded by $O(\text{dist}(q,\pi(q)))$. Let
$E_{1}^{\prime}=\\{\text{dist}(\cdot,L)\geq\epsilon^{1/4n}\\}\subset
L^{\prime}$, then $E_{1}^{\prime}$ has measure bounded by $C\epsilon^{1/4n}$
by Lemma 2.7 and (6).

Since $\pi:L^{\prime}\to L$ has degree one, it is surjective. Let $E$ be the
subset of $L$ with at least two preimages. The volume almost decreasing
property of $d\pi$, together with the assumption $|\theta|\leq\epsilon$ on
$L$, imply

$2\int_{E}\text{Re}\Omega=2\int_{E}e^{-\rho}dvol(1+O(\epsilon))\leq(1+O(\epsilon^{1/4n^{2}}))\int_{\pi^{-1}(E)}e^{-\rho}dvol.$

In particular

$\text{Vol}(E)+\text{Vol}(\pi^{-1}(E))\leq
C\int_{\pi^{-1}(E)}e^{-\rho}dvol\leq
C^{\prime}(\int_{\pi^{-1}(E)}e^{-\rho}dvol-\int_{E}\text{Re}\Omega).$ (9)

On the other hand, at the points $q\in L^{\prime}\setminus\pi^{-1}(E)\cup
E_{1}^{\prime}$, the difference between the tangent planes is bounded by

$|\vec{T}(q)-\vec{T}(\pi(q))|^{2}\leq
C(1-|d\pi(\vec{T})|+O(\epsilon^{1/4n})).$

Thus

$\begin{split}&\int_{L^{\prime}\setminus\pi^{-1}(E)}|\vec{T}(q)-\vec{T}(\pi(q))|^{2}dvol(q)\\\
\leq&C\epsilon^{1/4n}+C\int_{L^{\prime}\setminus\pi^{-1}(E)}(1-|d\pi(\vec{T})|)dvol\\\
\leq&C\epsilon^{1/4n}+C^{\prime}\int_{L^{\prime}\setminus\pi^{-1}(E)}(1-|d\pi(\vec{T})|)e^{-\rho}dvol.\end{split}$

Here the contribution from $E_{1}^{\prime}$ is absorbed into
$C\epsilon^{1/4n}$. From the pointwise inequality on
$L^{\prime}\setminus\pi^{-1}(E)\cup E_{1}^{\prime}$,

$|d\pi(\vec{T})|e^{-\rho}=(1-O(\epsilon^{1/4n}))|d\pi(\vec{T})||\text{Re}\Omega|_{L}|\geq\text{Re}\Omega(\pi_{*}\vec{T})(1-O(\epsilon^{1/4n})),$

the above is bounded by

$C\epsilon^{1/4n}+C^{\prime}(\int_{L^{\prime}\setminus\pi^{-1}(E)}e^{-\rho}dvol-\int_{L\setminus
E}\text{Re}\Omega).$

The coefficient $C^{\prime}$ can be arranged as the same (large) constant
appearing in (9). Adding the two contributions, and envoking the weighted
volume upper bound in Lemma 2.7,

$\begin{split}&\text{Vol}(E)+\text{Vol}(\pi^{-1}(E))+\int_{L^{\prime}\setminus\pi^{-1}(E)}|\vec{T}(q)-\vec{T}(\pi(q))|^{2}dvol(q)\\\
\leq&C\epsilon^{1/4n}+C^{\prime}(\int_{L^{\prime}}e^{-\rho}dvol-\int_{L}\text{Re}\Omega)\\\
\leq&C\epsilon^{1/4n}+C^{\prime}(\int_{L^{\prime}}\text{Re}\Omega-\int_{L}\text{Re}\Omega)=C\epsilon^{1/4n},\end{split}$

as required. ∎

######  Remark 10.

If $L$ is immersed instead of embedded, then we include the
$O(\epsilon^{1/4n^{2}})$ neighbourhood of the self intersections into $E$, and
the rest of the arguments run almost verbatim.

Recall the distance between two varifolds $L,L^{\prime}$ is measured by
F-metric

$\sup\\{|\int_{L}f(p,T_{p}L)dvol_{L}-\int_{L^{\prime}}f(q,T_{q}L^{\prime})dvol_{L^{\prime}}|\\}.$

where $f$ ranges over all functions on the Grassmannian bundle of
$n$-dimensional tangent subspaces over $X$, with $|f|\leq 1$ and Lipschitz
constant at most one. Given a uniform upper bound on the mass, the $F$-metric
induces the same topology as the weak topology on varifolds (because
continuous functions can be approximated by Lipschitz functions).

######  Corollary 2.12.

In the setting above, the $F$-metric between $L,L^{\prime}$ is bounded by
$C\epsilon^{1/8n}$.

###### Proof.

Since $|f|\leq 1$, the integrals on $E,E_{1}^{\prime}$ and $\pi^{-1}(E)$ are
bounded by $C\epsilon^{1/4n}$. At $q\in L^{\prime}\setminus\pi^{-1}(E)\cup
E_{1}^{\prime}$, by the Lipschitz bound on $f$,

$|f(q,T_{q}L^{\prime})-f(\pi(q),T_{\pi(q)}L)|\leq|q-\pi(q)|+|\vec{T}(q)-\vec{T}(\pi(q))|\leq
C\epsilon^{1/4n}+|\vec{T}(q)-\vec{T}(\pi(q))|.$

The discrepancy between the volume forms $\pi^{*}dvol_{L}$ and
$dvol_{L^{\prime}}$ is bounded by
$C|\vec{T}(q)-\vec{T}(\pi(q))|^{2}dvol_{L^{\prime}}$. Consquently, the
F-metric is bounded by

$C\epsilon^{1/4n}+\int_{L^{\prime}\setminus\pi^{-1}(E)}|\vec{T}(q)-\vec{T}(\pi(q))|dvol_{L^{\prime}}+C\int_{L^{\prime}\setminus\pi^{-1}(E)}|\vec{T}(q)-\vec{T}(\pi(q))|^{2}dvol_{L^{\prime}}.$

Using Prop. 2.11 and Cauchy-Schwarz, this is bounded by $C\epsilon^{1/8n}$. ∎

######  Acknowledgement.

The author is currently a Clay Research Fellow based at MIT. He thanks P.
Seidel for discussions.

## References

  * [1] Mohammed Abouzaid, Yohsuke Imagi. Nearby Special Lagrangians. arXiv:2112.10385.
  * [2] Akaho, Manabu; Joyce, Dominic. Immersed Lagrangian Floer theory. J. Differential Geom. 86 (2010), no. 3, 381–500.
  * [3] Almgren, F. Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35 (1986), no. 3, 451–547.
  * [4] Arnol’d, V. I. Dynamics of complexity of intersections. Bol. Soc. Brasil. Mat. (N.S.) 21 (1990), no. 1, 1–10.
  * [5] Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru. Lagrangian intersection Floer theory: anomaly and obstruction. Part I. AMS/IP Studies in Advanced Mathematics, 46.1. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. xii+396 pp.
  * [6] Li, Yang. Thomas-Yau conjecture and holomorphic curves. arXiv:2203.01467.
  * [7] Imagi, Yohsuke; Joyce, Dominic; Oliveira dos Santos, Joana. Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in $\mathbb{C}^{m}$. Duke Math. J. 165 (2016), no. 5, 847–933.
  * [8] Imagi, Yohsuke. A uniqueness theorem for gluing calibrated submanifolds. Comm. Anal. Geom. 23 (2015), no. 4, 691–715.
  * [9] Morgan, Frank. Geometric measure theory. A beginner’s guide. Fourth edition. Elsevier/Academic Press, Amsterdam, 2009. viii+249 pp.
  * [10] Neves, André. Recent progress on singularities of Lagrangian mean curvature flow. Surveys in geometric analysis and relativity, 413–438, Adv. Lect. Math. (ALM), 20, Int. Press, Somerville, MA, 2011.
  * [11] Seidel, Paul. Categorical dynamics. available on Seidel’s webpage.
  * [12] Thomas, R. P. Moment maps, monodromy and mirror manifolds. Symplectic geometry and mirror symmetry (Seoul, 2000), 467–498, World Sci. Publ., River Edge, NJ, 2001.
  * [13] Thomas, R. P.; Yau, S.-T. Special Lagrangians, stable bundles and mean curvature flow. Comm. Anal. Geom. 10 (2002), no. 5, 1075–1113.


# Signal Quality Auditing for Time-series Data

Chufan Gao1 2, Nicholas Gisolfi1, Artur Dubrawski1

###### Abstract

Signal quality assessment (SQA) is required for monitoring the reliability of
data acquisition systems, especially in AI-driven Predictive Maintenance (PMx)
application contexts. SQA is vital for addressing ”silent failures” of data
acquisition hardware and software, which when unnoticed, misinform the users
of data, creating the risk for incorrect decisions with unintended or even
catastrophic consequences. We have developed an open-source software
implementation of signal quality indices (SQIs) for the analysis of time-
series data. We codify a range of SQIs, demonstrate them using established
benchmark data, and show that they can be effective for signal quality
assessment. We also study alternative approaches to denoising time-series data
in an attempt to improve the quality of the already degraded signal, and
evaluate them empirically on relevant real-world data. To our knowledge, our
software toolkit is the first to provide an open source implementation of a
broad range of signal quality assessment and improvement techniques validated
on publicly available benchmark data for ease of reproducibility. The
generality of our framework can be easily extended to assessing reliability of
arbitrary time-series measurements in complex systems, especially when
morphological patterns of the waveform shapes and signal periodicity are of
key interest in downstream analyses.

## Introduction

Signal quality assessment (SQA) of time-series data is an important tool for
monitoring reliability of data acquisition systems. It is vital for
situational awareness of the maintainers and users of systems that rely on
such data. Silent failures of data acquisition software and hardware may cause
confusion in the consumers of the information being collected, and this
confusion may then lead to untoward or even catastrophic consequences.

The quality of an AI system is to a large extent a function of the quality of
the data used to train data-driven models as well as the data quality of
queries submitted for processing, and time-series data signals are often
corrupted with domain-specific artifacts which can introduce structured noise
that biases the learned decision logic of an AI system, while also distorting
its inferences. Examples of artifacts that often manifest in, e.g., healthcare
time series data include baseline wander, muscle artifacts, powerline
interference, or equipment failure (Satija, Ramkumar, and Manikandan 2018).
These artifacts can lead to critical, potentially deadly, errors in clinical
practice; similar negative outcomes can manifest when performing prognostic
analyses of engineered systems. Affected application areas include false alarm
reduction (Hravnak et al. 2016), physiological pattern discovery (Gao et al.
2019, 2021), determining cardiovascular sufficiency (Li, Pinsky, and Dubrawski
2022; Nagpal et al. 2019), and more. Issues with time-series data quality
represent a major hurdle to the widespread integration of Artificial
Intelligence into the PMx paradigm because poor data quality can significantly
reduce the accuracy and applicability of machine learning algorithms.

To illustrate our concepts, we focus on signal quality of clinically relevant
data to leverage relatively rich prior work in that domain. For instance,
existing reference databases of electrocardiogram (ECG) data are excellent
proxies for developing signal quality methods for broader application in the
scenarios where the subjects of analyses are physical systems instead of human
bodies.

While some work has been done in the area of open-source signal quality
analysis, there are very few open source implementations of SQA tools for
general use. For example,
Neurokit2111github.com/neuropsychology/NeuroKit(Makowski et al. 2021) \- one
of the largest ECG analysis libraries - only implements two signal quality
indices, one of which does not work well in practice. Aura
Healthcare222github.com/Aura-healthcare/ecg_q \- another ECG analysis software
repository - includes methods that overlap with Neurokit2. Neither of these
collections offer benchmark results computed on publicly available reference
data to enable their independent verification.

We develop and analyze an open-source, reproducible, python implementation of
many common SQA methods, including signal quality indices for analysis of
electrocardiogram (ECG), plethysmography (Pleth), and other vital sign
timeseries data prevalent in healthcare applications. Additionally, we
validate our results on both a closed-source bleeding dataset from a
healthcare research organization (Namas et al. 2009), and publicly available
datasets such as the MIT-BIH Arrhythmia database (Moody and Mark 2001).

The contributions of this work include:

  * •

Codifying core concepts in the benchmark analysis of Physionet 2011 ECG
Quality Classification Challenge (Silva, Moody, and Celi 2011). We show a new
result–a combination of all methods yields better performance than any
individual method alone.

  * •

Showing that improvements to data by computing and leveraging SQIs can lead to
improvements in downstream AI model performance. To demonstrate this, we
consider a clinical alert adjudication task of determining whether the bedside
monitor alert is due to real biological system status change, or to sensor
artifacts.

  * •

Considering SQIs in AI processing pipelines can reduce false detection and
false negative rates in anomaly detection settings, when the goal is to
identify subjects whose vital sign measurements look different from the rest
of a cohort.

  * •

Facilitating time-series signal denoising and present experimental results
with both quantitative and qualitative improvements.

  * •

Publicly releasing a software code repository for general use and reproduction
of experimental results.

## Methodology and Experiments

Descriptions of the methods, baselines, datasets and empirical results for
four research questions relating to signal quality auditing for time-series
data are presented here.

  1. 1.

Which SQI methods yield top-performing models in signal quality
classification?

  2. 2.

Which model classes benefit most from SQIs in an outlier detection task?

  3. 3.

How useful are SQIs at denoising time-series?

  4. 4.

What is the effect of ECG Denoising on Real vs Artifactual Alert Detection?

### Q1: Which SQI methods yield top-performing models in signal quality
classification?

Method | Signal Quality Indices (SQIs)
---|---
(1) Li et al. 2007 | b_sqi, p_sqi, k_sqi
(2) Clifford et al. 2012 |  | b_sqi, p_sqi, k_sqi, s_sqi, f_sqi,
---
bas_sqi
(3) Behar et al. 2013 | k_sqi, s_sqi, p_sqi, b_sqi
(4) Li et al. 2014 |  | b_sqi, p_sqi, k_sqi, s_sqi, f_sqi,
---
bas_sqi, bs_sqi, e_sqi, hf_sqi,
pur_sqi, rsd_sqi, ent_sqi
| (5) Geometric
---
Features
| Median, IQR, and Slope of HR.
---
R-peak interval STD. Samp. entr.,
appx. entr., Relative power,
and ratio of LF and HF bands
| (6) Average QRS
---
(Neurokit)
averageQRS_sqi
| (7) Zhao et al. 2018
---
(Neurokit)
zhao2018_sqi
| (8) Orphanidou et al.
---
2015
orphanidou2015_sqi
(9) Combined |  | All of the previously mentioned
---
SQIs and features
Table 1: We compare 9 methods shown in the table, with their corresponding SQI
features. Methods (6), (7) , and (8) each yield a single SQI feature.

We benchmarked a total of 8 prior works as well as a method that combines SQIs
from all those 8 prior research threads. (Li, Mark, and Clifford 2008) was one
of the first to apply machine learning to address ECG signal quality and
introduced a few common ECG SQIs: a beat-agreement calculated from two beat-
detection algorithms (bSQI), an inter-channel signal quality metric (iSQI),
the kurtosis of the signal (kSQI), and the spectral distribution of ECG
(pSQI). (Clifford et al. 2012) added 3 more SQIs, including skewness SQI
(sSQI), a flatness SQI (fSQI), and a feature to measure the relative power in
the baseline (basSQI). (Li, Rajagopalan, and Clifford 2014) introduced 6 more
SQIs: a baseline wander check in time domain (bsSQI), the relative energy in
the QRS complex (eSQI), the relative amplitude of high frequency noise
(hfSQI), the signal purity (purSQI), the relative standard deviation (STD) of
QRS complex (rsdSQI), the sample entropy (entSQI), the high frequency mask of
the ECG waveform (hfMSQI), and a periodic component analysis periodicity
measure (PiCASQI). (Behar et al. 2013) introduced an SQI based on the most
significant principal components obtained by principal component analysis
(pcaSQI). Orphanidou et al. proposed a SQI based on average correlation
between each beat and its average template. Neurokit2 (Makowski et al. 2021)
itself implemented an SQI that calculates a correlation of the QRS waveform to
the average QRS waveform (averageqrsSQI). Neurokit2 also implemented another
method: Zhao et al. (Zhao and Zhang 2018), which used a combination qSQI,
pSQI, kSQI, basSQ for a rule-based quality classification. Finally, we also
compared geometric shape features calculated from ECG including: median,
interquartile range (IQR), and the slope of heart rate (HR); r-peak interval
standard deviation; sample entropy; approximate entropy; relative power; and
ratio of low-frequency and high-frequency bands of the power spectrum of the
ECG. The specific implementations of each method can be found in our code.

We applied these methods to the original Physionet 2011 ECG signal quality
classification challenge, also known as the PICC dataset (Silva, Moody, and
Celi 2011). This dataset included ten-second annotated recordings of twelve-
lead ECGs, consisting of standard 12-lead ECG recordings, recorded
simultaneously for a minimum of 10 seconds at 500 Hz. ECG were annotated by
nurses, technicians, and volunteers, where 3 to 17 annotators independently
examined each ECG, to obtain a binary ”noisy” or ”not noisy” classification.
Additionally, this dataset was also used in an open classification challenge–
Table 2 shows the results of the challenge on the portion of the challenge
that required open source code (the most relevant to our work).

Participant | Accuracy
---|---
Xiaopeng Zhao | 0.914
Benjamin Moody | 0.896
Lars Johannesen | 0.880
Philip Langley | 0.868
Dieter Hayn | 0.834
Václav Chudáček | 0.833
Unofficial entries
George Moody | 0.894
Ikaro Silva | 0.802
Table 2: The PhysioNet/Computing in Cardiology Challenge 2011 Official
Results: Event 2 (open-source, open data set B), the most comparable event to
our case. However, this is still not exactly a proper comparison, since we do
not have access to data Set B (only Set A).

We used 5-fold group split cross-validation on the subjects in Set-A of the
dataset, so subject data do not overlap in training and testing in any
iteration. That yieldED 1996 ECG time-series for training and testing (50%
split), each 10 seconds long, recorded at 500 Hz. These raw time-series
signals are then featurized via the SQIs mentioned above. This yieldED a
straightforward binary classification: 0=not noisy, 1=noisy. A Random Forest
classifier was further trained on these SQI attributes.

Method | AUC | Accuracy
---|---|---
li2007 | 0.793 ± 0.033 | 0.869 ± 0.013
clifford2012 | 0.829 ± 0.041 | 0.877 ± 0.014
behar2013 | 0.803 ± 0.035 | 0.869 ± 0.013
li2014 | 0.830 ± 0.041 | 0.882 ± 0.013
orphanidou2015 | 0.826 ± 0.028 | 0.863 ± 0.012
averageqrs | 0.700 ± 0.025 | 0.858 ± 0.013
zhao2018 | 0.549 ± 0.013 | 0.792 ± 0.002
geometric | 0.807 ± 0.028 | 0.875 ± 0.013
all | 0.835 ± 0.033 | 0.888 ± 0.012
Table 3: Table of AUCs and Accuracies (with standard deviations) of 8 methods
on subject-based 5-fold cross validation and Random Forest with only single-
lead features. A comination of all methods perform the best (bolded).

The results in Table 3 showed that using all SQI features is the best in both
Area Under the Receiver Operating Characteristic (AUC) and classification
accuracy. We also note that the features from (Li, Rajagopalan, and Clifford
2014) performed close to all methods, indicating that their proposed SQIs (See
Table 1) were highly informative in this task. Additionally, we saw that,
compared to the original PhysioNet/Computing in Cardiology Challenge 2011
results in Table 2, the accuracy performance of the random forest ran on
single-lead features was comparable to the 3rd spot on the leaderboard as
shown in Table 3, suggesting that we were able to reproduce baseline results.
Furthermore, we saw that Neurokit2’s implementation of (Zhao and Zhang 2018)
does not fare well under actual evaluation. This highlights the effect of
conducting evaluation on real-world benchmark data, and that using all
features, including all SQIs and ECG features, appears to perform best.

### Q2: Which model classes benefit most from SQIs in an outlier detection
task?

In contrast to the previous question, here we considered the inherent ability
of SQIs as outlier detection features, rather than explicit classification.
Four popular Outlier Detection methods were evaluated for automatical
detection of noisy signals. First, K-Nearest Neighbor (Ramaswamy, Rastogi, and
Shim 2000) is a classic method of detecting outliers that is based on the
distance of a point from its kth-nearest neighbor. We can think of the outlier
score as the distance to a sample’s kth-nearest neighbor. Second, the
Isolation Forest algorithm (Liu, Ting, and Zhou 2008) seeks to separate
anomalous observations from other observations by computing the average number
of random splits needed for an observation to reach leaf nodes of trees
obtained by splitting data on randomly chosen features. The outlier score is
the number of splittings required for a sample to reach a leaf, averaged over
the forest. A One-class Support Vector Machine (SVM) (Schölkopf et al. 2001)
can be thought of as a model that is fit to the data, where anomalous data is
not well predicted by it. Fourth, AutoEncoders (AEs) (Aggarwal 2017) are
neural networks that map data onto low-dimensional embedding space, and then
reconstruct data from the respective embedding. Anomalies should yield higher
reconstruction errors as the model is not trained on them as much as on
regular data.

We trained each of these models on the PICC dataset as before, with the same
5-fold cross-validation strategy to ensure generalizability of our models. For
implementation of these algorithms, we followed the example in the open-
source, anomaly detection Python package PyOD333pyod.readthedocs.io. PyOD
implemented 30 anomaly detection algorithms, with evaluation results on 55
benchmark datasets. We instantiated each model as in the example code, trained
on the train split of the PICC dataset, and then evaluated on the test split.
Each model is trained on a mix of clean and noisy signals, with the percentage
of noisy signals in the dataset being known. This is allows each model to find
its own threshold for anomaly classification. Finally, each model was tested
to see if its predicted outliers match a ”noisy” classification.

Method | Test AUC | Test Accuracy
---|---|---
KNN | 0.396 ± 0.013 | 0.693 ± 0.003
IForest | 0.830 ± 0.006 | 0.844 ± 0.004
OCSVM | 0.492 ± 0.062 | 0.646 ± 0.015
AE | 0.819 ± 0.007 | 0.843 ± 0.003
Table 4: Results (with standard deviations) for the four outlier detection
algorithms. IForest performs the best (bolded).

From the results in Table 4 we see that Autoencoders and Isolation Forest
perform best in both metrics. In fact, the performance of Isolation Forest
would have ranked 5th in the actual competition, despite the fact that it is
only using information from the SQIs for outlier detection, not explicit
classification. This indicates that the proposed SQIs are reflective of the
true signal quality, which is promising for generalization to other signal
quality assessment tasks.

### Q3: How useful are SQIs at denoising time-series?

Figure 1: ECG Denoising Results example from the CNN autoencoder (the best
model in terms of Mean Squared Error). The blue is the original signal, and
the orange is the cleaned signal.

For this task, we tested three different methods for signal denoising at
varying levels of signal quality. Wavelet denoising (Donoho 1995) is an
established method of denoising signals by discrete wavelet decomposition
(DWT), thresholding the decomposed coefficients, and inverse wavelet
decomposition (IDWT). (Alfaouri and Daqrouq 2008) presents an approach based
on the threshold value of ECG signal determination using Wavelet Transform
coefficients that applies thresholding on the decomposed coefficients. For ECG
denoising, it is common to use the Daubechies-4 (db4) wavelet.

Empirical Mode Decomposition (EMD) Denoising (Flandrin, Rilling, and Goncalves
2004) is a data-driven decomposition method that does not require any basis,
unlike DWT. (Weng, Blanco-Velasco, and Barner 2006) proposed an ECG denoising
method based on Empirical Mode Decomposition, which is able to remove high
frequency noise with minimum signal distortion. The goal of EMD is to
decompose the signal into a sum of Intrinsic Mode Functions (IMFs), which are
like DWT coefficients. An IMF is defined as 1) function with equal number of
extrema and zero crossings (or at most differed by one) and 2) The envelopes
connecting the local minima and local maxima of an IMF have a local average of
zero (Weng, Blanco-Velasco, and Barner 2006). The process of extracting IMFs
from a given input signal is called the ”sifting process”. CNN Denoising
Autoencoders denoise signals by data compression and reconstruction. (Chiang
et al. 2019) introduced a wavelet denoising method using convolutional
denoising autoencoders (AE) for ECG.

Noise | dB | Wavelet MSE | EMD MSE | AE MSE
---|---|---|---|---
em | -6 | 0.0824 | 0.0684 | 0.0187
em | 0 | 0.0527 | 0.0483 | 0.0104
em | 6 | 0.0276 | 0.0337 | 0.0067
em | 12 | 0.0101 | 0.0213 | 0.0055
em | 18 | 0.0028 | 0.0171 | 0.0048
em | 24 | 0.0008 | 0.0177 | 0.0046
ma | -6 | 0.0839 | 0.0853 | 0.0109
ma | 0 | 0.0612 | 0.0681 | 0.0076
ma | 6 | 0.0249 | 0.0370 | 0.0063
ma | 12 | 0.0097 | 0.0246 | 0.0054
ma | 18 | 0.0026 | 0.0187 | 0.0047
ma | 24 | 0.0010 | 0.0189 | 0.0046
bw | -6 | 0.0794 | 0.0813 | 0.0063
bw | 0 | 0.0608 | 0.0655 | 0.0060
bw | 6 | 0.0378 | 0.0468 | 0.0057
bw | 12 | 0.0178 | 0.0296 | 0.0057
bw | 18 | 0.0077 | 0.0221 | 0.0045
bw | 24 | 0.0012 | 0.0188 | 0.0051
gn | -6 | 0.0885 | 0.0941 | 0.0272
gn | 0 | 0.0437 | 0.0549 | 0.0144
gn | 6 | 0.0221 | 0.0364 | 0.0066
gn | 12 | 0.0066 | 0.0212 | 0.0054
gn | 18 | 0.0026 | 0.0184 | 0.0047
gn | 24 | 0.0011 | 0.0174 | 0.0044
all | -6 | 0.0653 | 0.0685 | 0.0229
all | 0 | 0.0487 | 0.0566 | 0.0105
all | 6 | 0.0292 | 0.0412 | 0.0065
all | 12 | 0.0113 | 0.0239 | 0.0055
all | 18 | 0.0035 | 0.0187 | 0.0048
all | 24 | 0.0011 | 0.0171 | 0.0045
Table 5: Mean Squared Error of the predicted output ECG vs the actual clean
ECG for all 3 compared methods, at each type of noise and decibel. Bolded
results indicate the best for that level and type of noise. Figure 2: Plots of
Receiver Operating Characteristic (ROC) curves (with AUC and stanrdard
deviation) for Heart Rate alerts, with and without additional SQIs calculated
on raw (represented by the blue line) and denoised ECG (represented by the
orange line).

We used 2 public datasets to test our denoising methods. MIT-BIH arrhythmia
and MIT-BIH arrhythmia noise stress test databases. These datasets will be
used to construct ground truth noisy and clean ECG signals for evaluation. The
MIT-BIH Arrhythmia Database (Moody and Mark 2001; Mark et al. 1982; Moody and
Mark 1990) is an open access dataset available on PhysioNet (Goldberger et al.
2000) which contains 48 half-hour two-channel ambulatory ECG recordings, from
47 subjects. The recordings were digitized at 360 hz per channel with 11-bit
resolution over a 10 mV range. Two or more cardiologists independently
annotated almost every QRS signal. Disagreements were resolved to obtain
reference annotations for each beat (approximately 110,000 annotations total).
The MIT-BIH Noise Stress Test Database (NSTDB) (Moody, Muldrow, and Mark 1984)
is another database available on PhysioNet (Goldberger et al. 2000) which
includes 12 half-hour ECG recordings and 3 half-hour recordings of noise
typical in ambulatory ECG recordings. The three noise records were assembled
from ECG signals that contained predominantly baseline wander (bw), muscle
(EMG) artifact (ma), and electrode motion artifact (em) noise.

We manually added noise from the MIT-BIH dataset to control the type and
variety of noise added. For 2 clean records in the MIT-BIH dataset (118 and
119), we manually added 5 different types of noise (Electrode Motion artifact
(EM), Baseline Wander (BW), Muscle Artifact (MA)), Gaussian Noise (GN), as
well as a combination of all 4 previous noises at different levels of signal-
to-noise ratio (SNR) of (-6, 0, 6, 12, 18, 24). SNR is a metric used to
measure level of a desired signal to the level of background noise, where an
SNR of 0 dB indicates that the signal level is equal to the noise level. The
higher the ratio, the higher the signal level, and the better the signal
quality. All the noise was taken from the MIT-BIH noise stress test database
except Gaussian Noise, which was manually added. Finally, we splitted ECG
signals into windows of length 512, which is required by the autoencoder
denoising method. Splitting data into training and testing sets yielded 9408
samples for each and 18816 samples in total.

The full implementation details of the adapted methods are in the file
denoising.py in the released python package. The evaluation metric is Mean
Squared Error of the reconstructed signal vs the ground truth signal. For each
method, the input will be a noisy signal, and the output is expected to be a
cleaned version of that signal.

The results of the denoising is shown in Table 5. From the MSE table, we see
that Convolutional Autoencoder generally performed the best at denoising ECG
with low SNR (higher anounts of noise), whereas wavelet denoising performed
better for higher SNR (lower amounts of noise). It would be interesting to
investigate methods where wavelet denoising is used for higher SNR and
Convolutional Autoencoders are used for lower SNR.

Examples of noisy and denoised ECGs from the CNN autoencoder are shown in
Figure 1. From the figures, the cleaning process was able to at least remove
some artifacts that occur in the original signal. For example, in the first
figure, the baseline wander component was removed. Additionally, the CNN
Autoencoder was able to preserve the peak to peak alignment quite well.

However, there are still issues to be solved, as in the bottom right figure,
in cases of extreme noise, the algorithm may not be able to accurately denoise
the signal. The CNN Autoencoder looses information in the first 2/3 of the
time window. This could be related to hyperparameter choice (kernel size,
filters), and the strict requirement of the neural network for a fixed length
input time series. The final section addresses future work to tackle and
potentially solve these issues.

### Q4: What is the effect of ECG Denoising on Real vs Artifactual Alert
Detection?

For a real world predictive maintenance application, we tested the ability of
our methods to improve robustness of CRI alert sysems by improving
classification of real vs artifactual alert prediction. This is a binary
classification task, Real Alert=1 and Artifactual Alert=0. The CRI_Alert
dataset is not publicly available, but it is a good target for our anlysis. It
is a set of time series data from Intensive Care Unit patients from a
healthcare research organization. High rates of false alarms for
cardiorespiratory instability (CRI) in monitored patients cause alarm fatigue.
The CRI_Alert project was created to look at episodes of Cardio-Respiratory
Insufficiency (CRI) with two primary objectives: First, to distinguish between
real and artifactual CRI alerts; Second, to predict when a patient is likely
to have a CRI in the future. In this dataset, we considered CRI alerts based
on Heart Rate (HR). In terms of raw data, we have access to raw ECG time-
series, which are split into 6537 3-min windows, every 20 seconds,
corresponding to each alert. Since we have access to the raw ECG time-series
signals, we were able to featurize the signal into our SQIs. The raw ECG
signals were featurized by using all of the features as mentioned in Q1, as it
performed the best in terms of signal quality classification, and should
contains the most amount of information for the downstream task. This
featurization created a total of 22 features (all listed SQIs excluding
zhao2018_sqi). Additionally, 10 patient level random splits were used for
cross evaluation to ensure no overlap between patient data in the test-train
splits.

A random Random Forest was trained on the SQIs from two considered cases of
ECG signal. In the first case, the raw time-series was featurized as normal.
In the second case, a denoised version of the ECG was featurized instead. This
denoised version was obtained by resampling the raw ECG signal to a hz of 125
and processing it via the CNN Autoencoder.

From Figure 2, we see that the AUC performance of the Random Forest is
significantly better compared to the results ran on Raw ECG signals for Heart
Rate CRIs. This increase in performance indicates that the denoising algorithm
is working, as it is removing noise from clean signals, and failing to remove
noise from noisy ones, increasing the difference between the 2 classes. This
effect helps the Random Forest model more efficiently distinguish between real
and artifactual alerts, improving robustness and reducing downtime for
clinicians.

## Discussion and Conclusion

Data quality, and more specifically time-series signal quality, is a major
concern in in all real-world application contexts where measurement
uncertainty cannot be avoided. This concern is particularly felt in AI-driven
PMx for equipment, where the sensors responsible for measuring health of
components are, themselves, components that may fail in unexpected ways. To
mitigate risk of negative modeling outcomes, we studied common baseline
methods in Signal Quality Indices, Outlier detection, and Signal Denesoising.

Experiments that we performed dealt with clinically relevant data for
convenience during the development phase, where we wanted to obtain copious
amounts of publicly available data as well as numerous baselines in prior work
to compare against. This design choice was made due to challenges specific to
AI-driven PMx contexts. Many datasets are not publicly available, meaning that
there is no opportunity to compare to meaningful baselines as we did with the
PhysioNet Challenge. Our methods are general enough that they can be applied
in PMx contexts in a straightforward manner.

Our results show that SQIs are capable of reproducing 9 signal quality
classification methods, showing that a combination of all methods performs the
best in terms AUC and Accuracy. We showed that an Isolation Forest is able to
accurately determine whether a signal is noisy or not, showing that the SQIs
are an accurate representation of the true signal quality. In time-series
signal denoising appliations, our results show CNN Autoencoders performed the
best in terms of MSE when compared to two other state-of-the-art signal
denoising methods. Denoising significantly improved real-vs-artifact
detection. This demonstrates that there is a clear advantage in using SQIs as
well as denoising in terms of a concrete downstream, predictive maintenance
task of false alarm reduction. This conclusion in particular demonstrates the
benefit that we stand to gain by focusing on improving data quality in times-
series signals.

In terms of noisy signal classification, further work is required in terms of
characterizing the fault modes of the system. For example, investigating the
types of noise that is specifically causing the time-series signal to create
artifactual alerts would be vital in actively managing the reliability of
measurement equipment. Additionally, expansions to more traditional
applications of predictive maintenance, such as analyzing component vibration
time-series to maintain aircraft, pumps, and more are a logical next step.

Future work may also focus on different design choices. For example, the CNN
Autoencoder denoising model may be further tuned in terms of kernel size,
number of layers, and dimensionality. These features have a large impact in
model capability, so a more extensive evaluation of parameter choices is
needed. Grid search, Bayesian Optimization, or other techniques could be used
to search for the best parameter choices for the network. Furthermore, the
current model is limited with a fixed size requirement of the input signal;
future methods should seek to address this, perhaps with alignment via Dynamic
Time Warping, or by recent Neural Network methods that do not require a fixed
length input. Use of SQIs to measure signal quality in conjunction with the
denoising model may also allow the user to quantify the amount of expected
improvement.

We have released an open-source implementation
(github.com/chufangao/signal_quality) of all previously mentioned methods to
serve as an accessible, open source, toolkit for signal quality analysis. Our
hope is that such a tool for quickly implementing baselines will facilitate
more research in AI-driven PMx, specifically focusing on data quality
auditing.

## References

  * Aggarwal (2017) Aggarwal, C. C. 2017.  An introduction to outlier analysis.  In _Outlier analysis_ , 1–34. Springer.
  * Alfaouri and Daqrouq (2008) Alfaouri, M.; and Daqrouq, K. 2008.  ECG signal denoising by wavelet transform thresholding.  _American Journal of applied sciences_ , 5(3): 276–281.
  * Behar et al. (2013) Behar, J.; Oster, J.; Li, Q.; and Clifford, G. D. 2013.  ECG signal quality during arrhythmia and its application to false alarm reduction.  _IEEE transactions on biomedical engineering_ , 60(6): 1660–1666.
  * Chiang et al. (2019) Chiang, H.-T.; Hsieh, Y.-Y.; Fu, S.-W.; Hung, K.-H.; Tsao, Y.; and Chien, S.-Y. 2019\.  Noise reduction in ECG signals using fully convolutional denoising autoencoders.  _Ieee Access_ , 7: 60806–60813.
  * Clifford et al. (2012) Clifford, G.; Behar, J.; Li, Q.; and Rezek, I. 2012.  Signal quality indices and data fusion for determining clinical acceptability of electrocardiograms.  _Physiological measurement_ , 33(9): 1419.
  * Donoho (1995) Donoho, D. L. 1995.  De-noising by soft-thresholding.  _IEEE transactions on information theory_ , 41(3): 613–627.
  * Flandrin, Rilling, and Goncalves (2004) Flandrin, P.; Rilling, G.; and Goncalves, P. 2004.  Empirical mode decomposition as a filter bank.  _IEEE signal processing letters_ , 11(2): 112–114.
  * Gao et al. (2021) Gao, C.; Dubrawski, A.; Pinsky, M.; Clermont, G.; and Yoon, J. 2021.  Identification and Explanation of Severity of Bleeding-Induced Hypovolemia Using Unsupervised Deep Learning.  In _TP55. TP055 MECHANICAL VENTILATION, ICU MANAGEMENT, AND CV_ , A2841–A2841. American Thoracic Society.
  * Gao et al. (2019) Gao, C.; Falck, F.; Goswami, M.; Wertz, A.; Pinsky, M. R.; and Dubrawski, A. 2019\.  Detecting Patterns of Physiological Response to Hemodynamic Stress via Unsupervised Deep Learning.  _arXiv preprint arXiv:1911.05121_.
  * Goldberger et al. (2000) Goldberger, A. L.; Amaral, L. A.; Glass, L.; Hausdorff, J. M.; Ivanov, P. C.; Mark, R. G.; Mietus, J. E.; Moody, G. B.; Peng, C.-K.; and Stanley, H. E. 2000\.  PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals.  _circulation_ , 101(23): e215–e220.
  * Hravnak et al. (2016) Hravnak, M.; Chen, L.; Dubrawski, A.; Bose, E.; Clermont, G.; and Pinsky, M. R. 2016\.  Real alerts and artifact classification in archived multi-signal vital sign monitoring data: implications for mining big data.  _Journal of clinical monitoring and computing_ , 30(6): 875–888.
  * Li, Mark, and Clifford (2008) Li, Q.; Mark, R.; and Clifford, G. 2008.  Robust heart rate estimation from multiple asynchronous noisy sources using signal quality indices and a Kalman filter.  _Physiological measurement_ , 29(1): 15.
  * Li, Rajagopalan, and Clifford (2014) Li, Q.; Rajagopalan, C.; and Clifford, G. D. 2014.  A machine learning approach to multi-level ECG signal quality classification.  _Computer methods and programs in biomedicine_ , 117(3): 435–447.
  * Li, Pinsky, and Dubrawski (2022) Li, X.; Pinsky, M. R.; and Dubrawski, A. 2022.  Automated Assessment of Cardiovascular Sufficiency Using Non-Invasive Physiological Data.  _Sensors_ , 22(3): 1024.
  * Liu, Ting, and Zhou (2008) Liu, F. T.; Ting, K. M.; and Zhou, Z.-H. 2008.  Isolation forest.  In _2008 eighth ieee international conference on data mining_ , 413–422. IEEE.
  * Makowski et al. (2021) Makowski, D.; Pham, T.; Lau, Z. J.; Brammer, J. C.; Lespinasse, F.; Pham, H.; Schölzel, C.; and Chen, S. A. 2021.  NeuroKit2: A Python toolbox for neurophysiological signal processing.  _Behavior Research Methods_ , 1–8.
  * Mark et al. (1982) Mark, R.; Schluter, P.; Moody, G.; Devlin, P.; and Chernoff, D. 1982.  An annotated ECG database for evaluating arrhythmia detectors.  In _IEEE Transactions on Biomedical Engineering_ , volume 29, 600–600. IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC 345 E 47TH ST, NEW YORK, NY ….
  * Moody and Mark (1990) Moody, G. B.; and Mark, R. G. 1990.  The MIT-BIH arrhythmia database on CD-ROM and software for use with it.  In _[1990] Proceedings Computers in Cardiology_ , 185–188. IEEE.
  * Moody and Mark (2001) Moody, G. B.; and Mark, R. G. 2001.  The impact of the MIT-BIH arrhythmia database.  _IEEE Engineering in Medicine and Biology Magazine_ , 20(3): 45–50.
  * Moody, Muldrow, and Mark (1984) Moody, G. B.; Muldrow, W.; and Mark, R. G. 1984.  A noise stress test for arrhythmia detectors.  _Computers in cardiology_ , 11(3): 381–384.
  * Nagpal et al. (2019) Nagpal, C.; Li, X.; Pinsky, M. R.; and Dubrawski, A. 2019.  Dynamically Personalized Detection of Hemorrhage.  In _Machine Learning for Healthcare Conference_ , 109–123. PMLR.
  * Namas et al. (2009) Namas, R.; Ghuma, A.; Torres, A.; Polanco, P.; Gomez, H.; Barclay, D.; Gordon, L.; Zenker, S.; Kim, H. K.; Hermus, L.; et al. 2009.  An adequately robust early TNF-$\alpha$ response is a hallmark of survival following trauma/hemorrhage.  _PloS one_ , 4(12): e8406.
  * Ramaswamy, Rastogi, and Shim (2000) Ramaswamy, S.; Rastogi, R.; and Shim, K. 2000.  Efficient algorithms for mining outliers from large data sets.  In _Proceedings of the 2000 ACM SIGMOD international conference on Management of data_ , 427–438.
  * Satija, Ramkumar, and Manikandan (2018) Satija, U.; Ramkumar, B.; and Manikandan, M. S. 2018.  A review of signal processing techniques for electrocardiogram signal quality assessment.  _IEEE reviews in biomedical engineering_ , 11: 36–52.
  * Schölkopf et al. (2001) Schölkopf, B.; Platt, J. C.; Shawe-Taylor, J.; Smola, A. J.; and Williamson, R. C. 2001.  Estimating the support of a high-dimensional distribution.  _Neural computation_ , 13(7): 1443–1471.
  * Silva, Moody, and Celi (2011) Silva, I.; Moody, G. B.; and Celi, L. 2011.  Improving the quality of ECGs collected using mobile phones: The Physionet/Computing in Cardiology Challenge 2011.  In _2011 Computing in Cardiology_ , 273–276. IEEE.
  * Weng, Blanco-Velasco, and Barner (2006) Weng, B.; Blanco-Velasco, M.; and Barner, K. E. 2006.  ECG denoising based on the empirical mode decomposition.  In _2006 International Conference of the IEEE Engineering in Medicine and Biology Society_ , 1–4. IEEE.
  * Zhao and Zhang (2018) Zhao, Z.; and Zhang, Y. 2018.  SQI quality evaluation mechanism of single-lead ECG signal based on simple heuristic fusion and fuzzy comprehensive evaluation.  _Frontiers in Physiology_ , 9: 727.