Sigmoid functions and multiscale resolutionDaan Huybrechs and Lloyd N.
Trefethen

# Sigmoid functions and multiscale resolution of singularities ††thanks:
Submitted to the editors DATE.

Daan Huybrechs () Dept. of Computer Science, KU Leuven, 3001 Leuven, Belgium
<EMAIL_ADDRESS>Lloyd N. Trefethen () Mathematical Institute,
University of Oxford, Oxford OX4 4DY, UK<EMAIL_ADDRESS>
###### Abstract

In this short, conceptual paper we observe that essentially the same
mathematics applies in three contexts with disparate literatures: (1)
sigmoidal and RBF approximation of smooth functions, (2) rational
approximation of analytic functions near singularities, and (3) $hp$ mesh
refinement for solution of PDEs. The relationship of (1) and (2) is as simple
as the change of variables $s=\log(x)$, and our informal mnemonic for this
relationship is “sigmoid = log(ratapprox).”

###### keywords:

rational approximation, sigmoid function, logistic function, activation
function, radial basis function, $hp$ mesh refinement

41A20, 65D15, 68T07

##  1 Sigmoids $\leftrightarrow$ rational approximation

Functions with branch point singularities can be approximated with root-
exponential convergence by rational functions. For example, consider

(1) $f(x)=\sqrt{x},\quad x\in[\kern 0.4pt0,1].$

Since Donald Newman in 1964 [24] it has been known that there are degree $n$
rational functions

(2) $r_{n}(x)=a_{0}+\sum_{k=1}^{n}{a_{k}\over 1+x/\varepsilon_{k}}$

such that

(3) $\|f-r_{n}\kern 0.7pt\|=O(\exp(-C\sqrt{n}\kern 1.5pt)),\quad C>0,$

where $\|\cdot\|$ is the supremum norm on $[\kern 0.4pt0,1]$. What makes this
root-exponential convergence possible is that the poles
$\\{-\varepsilon_{k}\\}$ are exponentially clustered near the singularity at
$x=0$. This effect applies near any branch point singularity [12, 16, 33] of a
real or complex function and is the basis of “lightning PDE solvers” for the
Laplace, biharmonic, and Helmholtz equations in domains with corners [1, 5,
12, 13]. Figure 1 illustrates root-exponential convergence and exponential
clustering for this model problem.

Figure 1: Root-exponential convergence (left, $0\leq n\leq 20$) and
exponential clustering of poles near $0$ in $(-\infty,0)$ (right, $n=20$) for
degree $n$ minimax rational approximation of $f(x)=\sqrt{x}$ on $[\kern
0.4pt0,1]$. Note that the horizontal axis in the first plot is $\sqrt{n}$.
These effects generalize to rational approximation near any branch point
singularity.

The function $1/(1+x/\varepsilon_{k})$ of (2) is monotonically decreasing for
$x\in[\kern 0.4pt0,1]$ and takes values ${\approx}\,1$ for
$x\ll\varepsilon_{k}$ and ${\approx}\,0$ for $x\gg\varepsilon_{k}$. If we
introduce the change of variables

(4) $s=\log x\in[-\infty,0\kern 0.4pt],\quad x=e^{s}\in[\kern 0.4pt0,1],$

then this function is transformed into

(5) ${1\over 1+e^{s}/\varepsilon_{k}}={1\over 1+e^{s-s_{k}}},$

where $s_{k}=\log\varepsilon_{k}$. This function, or more properly its reverse
$1/(1+e^{s_{k}-s}),$ is the most basic example of a sigmoid function, known as
the logistic function. In physics it goes by the name of the Fermi or
Fermi–Dirac function, and it is an elementary transformation of the hyperbolic
tangent. Functions of this kind are prototypical activation functions in
neural networks, and the literature of this area is vast [7, 11, 17, 19, 26].

Figure 2: In the $s=\log x$ variable, the exponentially clustered poles of
Figure 1 become sigmoid functions $(\ref{act})$ translated to various center
points $s_{k}$. Root-exponential convergence of rational approximations
becomes a statement about approximation of smooth functions by linear
combinations of translates of a fixed smooth function.

With the change of variables (4), we may follow (1) and (2) to define

(6) $F(s)=f(x)=e^{s/2}$

and

(7) $R_{n}(s)=r_{n}(x)=a_{0}+\sum_{k=1}^{n}{a_{k}\over 1+e^{s-s_{k}}}.$

Equation (3) then implies that there are approximations (7) such that

(8) $\|F-R_{n}\|=O(\exp(-C\sqrt{n}\kern 1.5pt)),\quad C>0,$

where $\|\cdot\|$ is now the supremum norm on $[-\infty,0\kern 0.4pt]$.
Equation (8) is nothing else than a claim about root-exponential approximation
of a smooth function on $[-\infty,0\kern 0.4pt]$ by linear combinations of
translates of a standard smooth function, plus a constant. Following results
such as those of [9], this could probably be proved more simply in this
setting than in the setting of rational functions, where the standard proof
technique is the relatively advanced Hermite integral formula [32]. The root-
exponential rate results from balancing discretization errors associated with
separations $\Delta s_{k}=O(1/\sqrt{n}\kern 1.4pt)$ against truncation errors
associated with a grid extent $s_{\min{}}=\min s_{k}=O(\sqrt{n}\kern 1.4pt)$
[16]. Figure 2 illustrates the smooth functions in question for the problem of
Figure 1.

To summarize this opening section: with the change of variables $s=\log x$,
the approximation of a smooth function by linear combinations of translates of
a logistic function becomes equivalent to the approximation of a function with
a branch point singularity by rational functions with exponentially clustered
poles.

##  2 Rational approximation $\leftrightarrow$ $hp$ mesh refinement

Rational approximation with poles exponentially clustered near singularities
seems obviously akin to the resolution of functions near singularities by
exponentially refined meshes. Such techniques of mesh refinement are well
known in the literature of the finite element method and associated
approximation theory [8, 14, 15, 20, 27, 28].

Exponential clustering of poles is reflected in the approximately uniform
spacing on the semilogx scale in the right image of Figure 1, or equivalently,
the approximately uniform spacing of the sigmoid functions (5) in Figure 2.
However, it is notable that in both of these images the spacing is only
approximately uniform, growing sparser toward the left. This is the phenomenon
of tapered exponential clustering investigated in [33]. Quantitatively, one
finds that the density of poles with respect to the $s$ variable decreases
about linearly as $s$ decreases to some value $s_{\min{}}$. This distribution
brings a factor of $2$ improvement in convergence rate as a function of
$n$—because a uniform density would have the same convergence rate but twice
as many poles. (The more local sparsification in the rightmost few points of
Figures 1 and 2 is investigated in [16] with appeal to the asymptotic results
of Stahl [29].)

We have noticed that this tapered exponential clustering corresponds closely
to what is known as $hp$ mesh refinement ($h$ stands for grid spacing, $p$ for
order of approximation). In particular, the standard $hp$ mesh refinement
formula in one dimension has the same linear pattern just described, with
polynomial order taking the role of pole density. A singular function such as
$\sqrt{x}$ on $[\kern 0.4pt0,1]$ is approximated by piecewise polynomials on
intervals of lengths decreasing exponentially toward the singularity, with
polynomial representations of linearly decreasing degrees $\dots,3,2,1,0.$ It
is the same pattern, and it brings the same factor of 2 speedup for the same
reason.

One can explain linear tapering in various ways in various settings. In [33]
an argument is given based on potential theory. Here is an outline of the
simpler argument that originates with DeVore and Scherer [8, 27] in the study
of piecewise polynomial approximations of $x^{\alpha}$ on exponentially graded
meshes on $[\kern 0.4pt0,1]$. We speak for simplicity in terms of mesh
refinement by factors of $1/2$, though the optimal factor is actually
$(\sqrt{2}-1)^{2}\approx 0.172$. The observations of DeVore and Scherer were
generalized to ODE and PDE discretizations a few years later by Babuška and
his collaborators [14, 15]. In multiple dimensions, the details change.

  1. 1.

Approximation of $\sqrt{x}$ on $[1/2,1]$ is the same as approximation of
$\sqrt{2x}$ on $[1/4,1/2]$.

  2. 2.

Therefore approximation of $\sqrt{x}$ on $[1/4,1/2]$ is the same problem too,
but with an accuracy criterion loosened by a factor $\sqrt{2}$.

  3. 3.

Functions like these, bounded away from singularities, can be approximated by
polynomials with exponential convergence.

  4. 4.

Therefore, that loosening by the factor $\sqrt{2}$ allows one to lower the
degree of the polynomial by a constant increment and still get the same
accuracy.

  5. 5.

Repeat on $[1/8,1/4]$, $[1/16,1/8],\dots.$

To summarize this second section: the standard formula for $hp$ mesh
refinement in 1D involves a linear decrease of polynomial degree toward the
singularity, and this corresponds to the linear decrease of pole density on a
logarithmic scale in tapered exponential clustering of poles in rational
approximation, resulting in the same factor of 2 speedup.

We note that this standard $hp$ mesh refinement strategy is not the only way
to achieve linear tapering. An alternative would be to hold the polynomial
degree $p$ fixed and instead refine $h$ super-exponentially at the
singularity. We do not know if such a prescription has been used in finite
element calculations.

##  3 Double exponential and generalized Gauss quadrature

In the area of quadrature or numerical integration, many methods have been
developed for dealing with singularities. When an endpoint singularity like
$x^{\alpha}$ is known, a targeted quadrature formula can be derived: the
prototype is Gauss–Jacobi quadrature. For dealing with more complicated or
unknown singularities, however, more general techniques have been proposed.
One is double exponential or tanh-sinh quadrature [2, 23, 31]. As illustrated
in Figure 14 of [33], the tanh-sinh formula with standard parameter choices
produces a tapered exponentially clustered distribution of quadrature points,
indicating that it is probably related to what is seen with rational
approximation and $hp$ mesh refinement.

Kirill Serkh (private communication) has shown us that similar effects also
arise with generalized Gauss and universal quadrature formulas [3, 4]. These
are quadrature formulas that are constructed by linear algebra methods related
to Gauss quadrature so as to be efficient at integrating not just a single
singularity such as a fixed power $x^{\alpha}$ but a range of singularities
such as $x^{\alpha}$, $\alpha\in[\kern 0.4pt0,1]$. Again it appears that in
important cases, the nodes are exponentially clustered near the singularity
with a tapered distribution.

##  4 The physics of $s=\log(x)$: separation of scales

The change of variables $s=\log x$ is not just an algebraic trick. It also has
a physical interpretation alluded to in section 5 of [33]; see for example
Figure 11 of that paper.

Even before introducing the change of variables, the point can be seen in the
$x$ variable. The function $1/(1+x/\varepsilon_{k})$, with its pole at
distance $\varepsilon_{k}$ to the left of $x=0$, is essentially constant and
hence inactive to the right of $x=0$ for $x\ll\varepsilon_{k}$ (taking the
value $1$) and $x\gg\varepsilon_{k}$ (taking the value $0$). It is only for
$x\approx\varepsilon_{k}$ that this function is active. Thus exponentially
separated poles $\\{-\varepsilon_{k}\\}\subseteq(-\infty,0)$ are physically
decoupled, operating in independent regimes, with each pole at
$-\varepsilon_{k}<0$ affecting the approximation on $(0,\infty)$ only for
$x\approx\varepsilon_{k}$.

Figure 3: Sketch of the “physics” of the change of variables $s=\log x$. This
is a conformal map of the upper half $x$-plane to an infinite strip in the
$s$-plane, with poles $-\varepsilon_{k}$ and sample locations
$\varepsilon_{k}$ on exponentially separated scales mapping to poles
$s_{k}+\pi i$ and sample locations $s_{k}$ on opposite sides of the strip. The
well-known exponential decay of influences along strips explains why a pole at
$s_{k}+\pi i$ has a significant effect at $s_{k}$ but a much smaller effect at
a different point $s_{j}$ with $|s_{j}-s_{k}|\gg 1$.

The change of variables $s=\log x$ suggests a physical explanation of this
separation of scales effect. A problem with a singularity at $x=0$ can be
motivated as a model of a corner singularity in a PDE problem. Specifically,
suppose a Laplace problem is posed in the upper half complex $x$-plane with a
singularity at $x=0$. Changing to $s=\log x$ transplants this problem to the
infinite strip $0<\hbox{Im\kern 0.5pt}s<\pi$ in the $s$-plane. The problem is
now smooth, with the singularity moved to $-\infty$. Exponentially clustered
poles $-\varepsilon_{k}\in(-\infty,0)$ become well separated poles $s_{k}+\pi
i$ on the upper side of the strip. And now, as sketched in Figure 3, it is a
well-known effect of potential theory (or elasticity, where it is called the
St.Venant principle) that influences decay exponentially with distance along a
strip.

The argument just made is tied to the Laplace equation, because its solutions
are invariant with respect to conformal maps. However, the essence of the
matter will be the same for any problem whose highest order derivative is the
Laplacian, because close to a singularity, this term will dominate. With the
Helmholtz equation $\Delta u+k^{2}u=0$, for example, the influence of the
$k^{2}u$ term quickly shuts off to zero as one comes exponentially close to a
corner.

##  5 Radial basis functions and other activation functions

A rational function $r(x)$ as in (2) is a sum of simple poles, which in the
$s$ variable becomes a linear combination of sigmoids as in Figure 2. The
picture changes little for various other activation functions. For example, if
the poles $(x+\varepsilon_{k})^{-1}$ in (2) are replaced by powers
$(x+\varepsilon_{k})^{-a}$ for an arbitrary $a>0$, Figure 2 does not change
very much. This matches theoretical and experimental results in neural
networks, where choices between activation functions are typically based more
on the efficiency of learning algorithms such as stochastic gradient descent
than on approximation power.

As many authors have noted, a closely related topic is that of approximation
by radial basis functions. Here again one approximates a complicated function
by a linear combination of translates of a simpler fixed function, and
convergence may be very fast when the latter is smooth [6, 9, 10, 21, 22, 25].

In closing we note that although sigmoidal and other activation functions are
important in neural networks and deep learning, the present paper touches only
the surface of that discipline. The approximation (7) is not composite, but
involves just what is conventionally called a single “hidden layer.” Smooth
activation functions have accuracy advantages for single-layer approximation,
but in the multi-layer setting of deep learning, that advantage diminishes and
the simpler non-smooth function known as ReLU is used more often [19, 30]. We
cannot resist mentioning that the universal approximation power of ReLU units
was exploited by Henri Lebesgue at age 23 in his first published paper, which
presented a new proof of the Weierstrass approximation theorem [18].

## Acknowledgments

We are grateful to Kirill Serkh of the University of Toronto for showing us
the tapered exponentially clustered nodes of generalized Gauss and universal
quadrature formulas.

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# Emergent anomaly of Fermi surfaces: a simple derivation from Weyl fermions

Ruochen Ma  Perimeter Institute for Theoretical Physics, Waterloo, Ontario,
Canada N2L 2Y5  Chong Wang  Perimeter Institute for Theoretical Physics,
Waterloo, Ontario, Canada N2L 2Y5

###### Abstract

The Fermi liquid has an emergent low-energy symmetry that corresponds to
charge conservation at each momentum on the Ferm surface. Recently Else,
Thorngren and Senthil (ETS) argued that the emergent symmetry had a specific
t’Hooft anomaly[1]. We give a simple derivation of the ETS anomaly using the
chiral anomaly of Weyl fermions.

The Landau Fermi liquid[2], along with certain types of non-Fermi liquids[3],
is characterized by a large emergent symmetry: the charge at each momentum on
the Fermi surface is conserved at low energy. Recently, Else, Thorngren and
Senthil (ETS) pointed out[1] that this large emergent symmetry, mathematically
known as the loop group $L^{d-1}U(1)$ where $d$ is the space dimension, has a
t’Hooft anomaly of the Chern-Simons form

$\displaystyle\frac{i}{(d+1)!(2\pi)^{d}}\int\mathcal{A}(d\mathcal{A})^{d},$
(1)

where the products in the integrand are the wedge product. The probe gauge
field $\mathcal{A}$ not only lives in the real spacetime coordinates, but also
in the $(d-1)$ dimensional momentum space on the Fermi surface – this is the
manifestation of the charge conservation at each Fermi surface momentum, so
the probe gauge field can couple to each momentum separately. As a t’Hooft
anomaly, Eq. (1) lives in one extra dimension, so the integration in total has
$(d+1)+(d-1)+1=2d+1$ dimensions. Many known properties of Fermi liquids (and
even certain non-Fermi liquids) are nicely unified under this anomaly,
including Luttinger theorem, quantum oscillation and anomalous Hall
conductivity.

The ETS anomaly Eq. (1) was proposed[1] based on the phase space
dimensionality (the Chern-Simons term being essentially the only known form)
and the fact that the anomaly reproduces several known properties of Fermi
liquids. In this note we give a simple direct derivation of the ETS anomaly.

For simplicity we derive the anomaly for the $(2+1)d$ Fermi surface (the
surface itself has one dimension), but our derivation is quite readily
generalized to higher dimensions.

We start from a single Weyl fermion in $(3+1)d$, coupled to an ordinary $U(1)$
gauge field $A$:

$\displaystyle
H=\sum_{i=x,y,z}v_{i}\psi^{\dagger}\sigma_{i}(i\partial_{i}-A_{i})\psi,$ (2)

where $v_{i}>0$ is the velocity of the Weyl fermion in the $i$-direction. This
theory has the famous chiral anomaly[4, 5], described by a Chern-Simons term
in $(3+1)+1=5$ dimensions:

$\displaystyle\frac{i}{6(2\pi)^{2}}\int AdAdA.$ (3)

Crucially, this anomaly does not change under various deformations of the
theory. In particular, consider turning on a uniform magnetic field in
$\hat{z}$ direction $B\hat{z}$. The important feature at low energy is the
chiral Landau level, which contains many ($N=\int dxdyB/2\pi$) branches of
chiral fermions moving in $\hat{z}$ direction. Now take the Landau gauge
$A_{x}=0,A_{y}=Bx$, then the momentum $k_{y}$ is a good quantum number that
can be used to label the chiral fermions. At low energy we obtain a one-
dimensional Fermi surface labeled by $k_{y}$, with a chiral fermion moving in
$\hat{z}$ direction for each momentum $k_{y}$. Assuming periodic boundary
condition in the $\hat{x}$ direction with linear size $L_{x}$, the momentum
$k_{y}$ takes value in $[0,BL_{x})$ with $k_{y}\sim k_{y}+BL_{x}$. We can then
interpret $k_{y}$ as parametrizing a circle, and the Fermi surface is
indistinguishable from a conventional one when observed at low energy. Namely,
all we have done is to find a slightly unconventional UV regularization of a
conventional Fermi surface.

This one-dimensional Fermi surface should of course inherit the original
chiral anomaly of the Weyl fermion Eq. (3). But what does the $x$ direction in
Eq. (3) mean for the Fermi surface (which has fermions labeled by $k_{y}$,
moving in $\hat{z}$ but no explicit $x$ dependence)? To understand this, we
take the single Landau level limit $v_{x}\to\infty$ and completely project the
physics to the chiral Landau levels111Of course in practice we do not need the
strict limit of $v_{x}\to\infty$. All that is needed is that all other energy
scales (temperature, frequency and interaction strength) are well below the
Landau gap $\sim Bv_{x}v_{y}$.. In this limit we can take the standard
substitution $x\to k_{y}/B$, $\partial_{x}\to B\partial_{k_{y}}$ and by gauge
invariance $A_{x}\to BA_{k_{y}}$. The chiral anomaly then becomes exactly the
ETS anomaly, where the integration in the $x$ coordinate in Eq. (3) becomes
that of the momentum coordinate in Eq. (1).

As pointed out in Ref. [1], the $A_{k_{y}}$ gauge field can be interpreted as
the Berry connection on the Fermi surface and contributes to the Hall
conductance[6]. We can verify this connection explicitly as follows. Consider
a configuration $A_{x}=f(x)\neq 0$, $A_{y}=Bx$, the equation of motion can
still be solved by chiral fermions labeled by $k_{y}$, with the same number of
degeneracy on the lowest Landau level. We then expect the Fermi surface Berry
phase to correspond to the magnetic flux through the $x$-circle:
$\Phi_{B}=\oint_{x}A_{x}$. The Hall conductance $\sigma_{yz}$ can be measured
from the Berry phase of the adiabatic evolution
$\phi_{-2\pi}^{y}\phi_{-2\pi}^{z}\phi_{2\pi}^{y}\phi_{2\pi}^{z}$, where
$\phi_{\pm 2\pi}^{\alpha}$ denotes an adiabatic $\pm 2\pi$ flux threading
through the $\alpha$-circle. The effect of $\phi_{\pm 2\pi}^{z}$ is to
increase (decrease) the chemical potential in $k_{z}$ by one minimum unit,
namely to increase (decrease) one filled Landau level in $xy$ plane. The Berry
phase of the evolution
$\phi_{-2\pi}^{y}\phi_{-2\pi}^{z}\phi_{2\pi}^{y}\phi_{2\pi}^{z}$ is therefore
equal to the Berry phase of the simple flux-threading $\phi_{2\pi}^{y}$ for a
single filled Landau level. The latter is simply given by
$\oint_{x}A_{x}=\Phi_{B}$, exactly what we anticipated.

We also note that when going from the Weyl chiral anomaly to the ETS anomaly,
we could subtract off the “background” magnetic field by making the
substitution $A_{y}\to A_{y}+Bx\sim A_{y}+k_{y}$. This results in an
additional piece that can be simply interpreted as the sum of the chiral
anomalies of $(1+1)d$ chiral fermions. This simple sum of $(1+1)d$ chiral
anomaly is always present and is not included in the ETS anomaly.

To recap, what we have done is to find a UV regularization – a Weyl fermion
under a magnetic field – of a Fermi surface, from which the IR anomaly follows
simply from the UV anomaly. The advantage of this regularizations is that the
conserved charge at each point on the Fermi surface (the generator of $LU(1)$)
is local (in $x$), in sharp contrast to usual settings in which the conserved
charge at a specific momentum $k_{y}$ is highly non-local in real space. The
locality of the generators allow them to be coupled to a probe gauge field,
which is fundamentally why the information of the $LU(1)$ ETS anomaly can be
encoded in the UV chiral anomaly.

We end with several additional remarks:

  1. 1.

The $(3+1)d$ Weyl fermion also comes with a mixed $U(1)$-gravity anomaly. This
anomaly becomes a mixed $LU(1)$-gravity anomaly for the Fermi surface. This
has a familiar interpretation: the Berry phase on the Fermi surface gives a
contribution to the thermal Hall conductance of the metal[6].

  2. 2.

It is rather straightforward to generalize our argument to higher dimensions:
for the Weyl fermion, we can increase the space dimension by $2$ (recall that
Weyl fermion exists in even spacetime dimensions), and put another magnetic
flux density in the two new dimensions. We will again have chiral Landau
levels (ultimately guaranteed by the chiral anomaly of the Weyl fermion).
Repeating our earlier argument for the two new dimensions, it is easy to see
that we have increased the Fermi surface dimension by one and converted the
other dimension into momentum. We can repeat this procedure arbitrarily to
access higher dimensions, and the ETS anomaly Eq. (1) follows simply from the
chiral anomaly of a single Weyl fermion in dimension $D=2d$.

  3. 3.

There is one curious feature when the Weyl fermion lives in dimension
$D=4n+2$, where a pure gravitational anomaly exists[7]. One natural question
is whether this pure gravitational anomaly has any implication for the Fermi
surface (in spacetime dimension $(2n+2)$). However, since the Fermi surface is
quite trivial when the $LU(1)$ is completely broken (the Fermi surface can
easily be gapped then), it seems unlikely that an analogue of pure
gravitational anomaly will play a serious role for the Fermi surface. We leave
a detailed analysis of this aspect to future work.

###### Acknowledgements.

We thank T. Senthil and Liujun Zou for illuminating discussions. We also thank
Anton Burkov, Alexander Hickey and Xuzhe Ying for discussions and for a
related collaboration. RM and CW acknowledge support from the Natural Sciences
and Engineering Research Council of Canada (NSERC) through a Discovery Grant.
Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science and Industry Canada and
by the Province of Ontario through the Ministry of Colleges and Universities.

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