arXiv subjects

Sasha Sodin

Publications and source records attributed to Sasha Sodin.

At least 19 recordsLinked to original sources

arXiv2026

Randomly initialized autoencoders: fixed points and edge-of-chaos

In this paper we study autoencoders, a special class of deep neural nets (DNNs) whose performance can be characterized via their fixed points. This perspective naturally raises questions of existence, stability, and basins of attraction of these fixed points. These questions are addressed via the contractive properties of autoencoders, and are closely related to the notion of edge-of-chaos. Edge-of-chaos (EoC) is an important notion in the theory of DNNs. It describes the critical regime separating ordered and chaotic signal propagation through a randomly initialized network. Initialization at or near this critical regime offers several theoretical and practical advantages, including stability of the network w.r.t. perturbations of the input. EoC was previously introduced for broad classes of neural networks using mean-field averaging methods. In this paper we modify the notion of EoC for the study of autoencoders. Specifically, we introduce local and global EoC for autoencoders that control local (small) and global (arbitrary) perturbations of the input respectively. The study of stability of autoencoders falls within the scope of nonlinear problems in Random Matrix Theory (RMT). Our analysis of local EoC is based on spectral techniques of RMT, whereas global EoC is studied by employing Sudakov-Fernique inequality for Gaussian processes.

cs.LG↗

arXiv2026

How weak are weak factors? Uniform inference for signal strength in signal plus noise models

The paper analyzes four classical signal-plus-noise models: the factor model, spiked sample covariance matrices, the sum of a Wigner matrix and a low-rank perturbation, and canonical correlation analysis with low-rank dependencies. The objective is to construct confidence intervals for the signal strength that are uniformly valid across all regimes - strong, weak, and critical signals. We demonstrate that traditional Gaussian approximations fail in the critical regime. Instead, we introduce a universal transitional distribution that enables valid inference across the entire spectrum of signal strengths. The approach is illustrated through applications in macroeconomics and finance.

stat.ME↗

arXiv2025

Irreducibility of the characteristic polynomials of random tridiagonal matrices

Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension is irreducible, with probability exponentially close to one; moreover, its Galois group over the rational numbers is either the symmetric or the alternating group. This is the counterpart of the results of Breuillard--Varjú (for polynomials with independent coefficients), and with those of Eberhard and Ferber--Jain--Sah--Sawhney (for full random matrices). We also analyse a related class of random tridiagonal matrices for which the Galois group is much smaller.

math.NT↗

arXiv2025

On the Hausdorff measure of sets of non-Lyapunov behaviour, and a Jarnik-type theorem for random Schroedinger operators

We consider the growth of the norms of transfer matrices of ergodic discrete Schrödinger operators in one dimension. It is known that the set of energies at which the rate of exponential growth is slower than prescribed by the Lyapunov exponent is residual in the part of the spectrum at which the Lyapunov exponent is positive. On the other hand, this exceptional set is of vanishing Hausdorff measure with respect to any gauge function $ρ(t)$ such that $ρ(t)/t$ is integrable at zero. Here we show that this condition on $ρ(t)$ can not in general be improved: for operators with independent, identically distributed potentials of sufficiently regular distribution, the set of energies at which the rate of exponential growth is arbitrarily slow has infinite Hausdorff measure with respect to any gauge function $ρ(t)$ such that $ρ(t)/t$ is non-increasing and not integrable at zero. The main technical ingredient, possibly of independent interest, is a Jarník-type theorem describing the Hausdorff measure of the set of real numbers well approximated by the eigenvalues of the Schrödinger operator. The proof of this result relies on the theory of Anderson localisation and on the mass transference principle of Beresnevich--Velani.

math.SP↗

arXiv2024

Integrability of pushforward measures by analytic maps

Given a map $ϕ:X\rightarrow Y$ between $F$-analytic manifolds over a local field $F$ of characteristic $0$, we introduce an invariant $ε_{\star}(ϕ)$ which quantifies the integrability of pushforwards of smooth compactly supported measures by $ϕ$. We further define a local version $ε_{\star}(ϕ,x)$ near $x\in X$. These invariants have a strong connection to the singularities of $ϕ$. When $Y$ is one-dimensional, we give an explicit formula for $ε_{\star}(ϕ,x)$, and show it is asymptotically equivalent to other known singularity invariants such as the $F$-log-canonical threshold $\operatorname{lct}_{F}(ϕ-ϕ(x);x)$ at $x$. In the general case, we show that $ε_{\star}(ϕ,x)$ is bounded from below by the $F$-log-canonical threshold $λ=\operatorname{lct}_{F}(\mathcal{J}_ϕ;x)$ of the Jacobian ideal $\mathcal{J}_ϕ$ near $x$. If $\dim Y=\dim X$, equality is attained. If $\dim Y<\dim X$, the inequality can be strict; however, for $F=\mathbb{C}$, we establish the upper bound $ε_{\star}(ϕ,x)\leqλ/(1-λ)$, whenever $λ<1$. Finally, we specialize to polynomial maps $φ:X\rightarrow Y$ between smooth algebraic $\mathbb{Q}$-varieties $X$ and $Y$. We geometrically characterize the condition that $ε_{\star}(φ_{F})=\infty$ over a large family of local fields, by showing it is equivalent to $φ$ being flat with fibers of semi-log-canonical singularities.

math.AG↗

arXiv2023

Sets of non-Lyapunov behaviour for scalar and matrix Schrödinger cocycles

We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schrödinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence $p_n$ there is a residual set of energies in the spectrum on which the middle singular value (the $W$-th out of $2W$) grows no faster than $p_n^{-1}$. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function $ρ(t)$ such that $ρ(t)/t$ is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each $k$, the average of the first $k$ Lyapunov exponents is the logarithmic potential of a probability measure.

math.SP↗

arXiv2022

The umpteen operator and its Lifshitz tails

As put forth by Kerov in the early 1990s and elucidated in subsequent works, numerous properties of Wigner random matrices are shared by certain linear maps playing an important rôle in the representation theory of the symmetric group. We introduce and study an operator of representation-theoretic origin which bears some similarity to discrete random Schrödinger operators acting on the $d$-dimensional lattice. In particular, we define its integrated density of states and prove that in dimension $d \geq 2$ it boasts Lifshitz tails similar to those of the Anderson model. The construction is closely related to an infinite-board version of the fifteen puzzle, a popular sliding puzzle from the XIX-th century. We estimate, using a new Peierls argument, the probability that the puzzle returns to its initial state after $n$ random moves. The Lifshitz tail is deduced using an identification of our random operator with the action of the adjacency matrix of the puzzle on a randomly chosen representation of the infinite symmetric group.

math-ph↗

arXiv2022

Anderson localisation for quasi-one-dimensional random operators

In 1990, Klein, Lacroix, and Speis proved (spectral) Anderson localisation for the Anderson model on the strip of width $W \geqslant 1$, allowing for singular distribution of the potential. Their proof employs multi-scale analysis, in addition to arguments from the theory of random matrix products (the case of regular distributions was handled earlier in the works of Goldsheid and Lacroix by other means). We give a proof of their result avoiding multi-scale analysis, and also extend it to the general quasi-one-dimensional model, allowing, in particular, random hopping. Furthermore, we prove a sharp bound on the eigenfunction correlator of the model, which implies exponential dynamical localisation and exponential decay of the Fermi projection. Our work generalises and complements the single-scale proofs of localisation in pure one dimension ($W=1$), recently found by Bucaj-Damanik-Fillman-Gerbuz-VandenBoom-Wang-Zhang, Jitomirskaya-Zhu, Gorodetski-Kleptsyn, and Rangamani.

math-ph↗

arXiv2022

Local tail bounds for polynomials on the discrete cube

Let $P$ be a polynomial of degree $d$ in independent Bernoulli random variables which has zero mean and unit variance. The Bonami hypercontractivity bound implies that the probability that $|P| > t$ decays exponentially in $t^{2/d}$. Confirming a conjecture of Keller and Klein, we prove a local version of this bound, providing an upper bound on the difference between the $e^{-r}$ and the $e^{-r-1}$ quantiles of $P$.

math.PR↗

arXiv2022

Lower bounds on Anderson-localised eigenfunctions on a strip

It is known that the eigenfunctions of a random Schrödinger operator on a strip decay exponentially, and that the rate of decay is not slower than prescribed by the slowest Lyapunov exponent. A variery of heuristic arguments suggest that no eigenfunction can decay faster than at this rate. We make a step towards this conjecture (in the case when the distribution of the potential is regular enough) by showing that, for each eigenfunction, the rate of exponential decay along any subsequence is strictly slower than the fastest Lyapunov exponent, and that there exists a subsequence along which it is equal to the slowest Lyapunov exponent.

math-ph↗

arXiv2021

Upper bounds on quantum dynamics in arbitrary dimension

Motivated by the research on upper bounds on the rate of quantum transport for one-dimensional operators, particularly, the recent works of Jitomirskaya--Liu and Jitomirskaya--Powell and the earlier ones of Damanik--Tcheremchantsev, we propose a method to prove similar bounds in arbitrary dimension. The method applies both to Schroedinger and to long-range operators. In the case of ergodic operators, one can use large deviation estimates for the Green function in finite volumes to verify the assumptions of our general theorem. Such estimates have been proved for numerous classes of quasiperiodic operators in one and higher dimension, starting from the works of Bourgain, Goldstein, and Schlag. One of the applications is a power-logarithmic bound on the quantum transport defined by a multidimensional discrete Schrödinger (or even long-range) operator associated with an irrational shift, valid for all Diophantine frequencies and uniformly for all phases. To the best of our knowledge, these are the first results on the quantum dynamics for quasiperiodic operators in dimension greater than one that do not require exclusion of a positive measure of phases. Moreover, and in contrast to localisation, the estimates are uniform in the phase. The arguments are also applicable to ergodic operators corresponding to other kinds of base dynamics, such as the skew-shift.

math-ph↗

arXiv2021

Power substitution in quasianalytic Carleman classes

Consider an equation of the form $f(x)=g(x^k)$, where $k>1$ and $f(x)$ is a function in a given Carleman class of smooth functions. For each $k$, we construct a Carleman-type class which contains all the smooth solutions $g(x)$ to such equations. We prove, under regularity assumptions, that if the original Carleman class is quasianalytic, then so is the new class. The results admit an extension to multivariate functions.

math.CA↗

arXiv2020

Anderson localisation in stationary ensembles of quasiperiodic operators

An ensemble of quasi-periodic discrete Schrödinger operators with an arbitrary number of basic frequencies is considered, in a lattice of arbitrary dimension, in which the hull function is a realisation of a stationary Gaussian process on the torus. We show that, for almost every element of the ensemble, the quasi-periodic operator boasts Anderson localization with simple pure point spectrum at strong coupling. One of the ingredients of the proof is a new lower bound on the interpolation error for stationary Gaussian processes on the torus (also known as local non-determinism).

math-ph↗

arXiv2020

On a uniqueness theorem of E. B. Vul

We recall a uniqueness theorem of E. B. Vul pertaining to a version of the cosine transform originating in spectral theory. Then we point out an application to the Bernstein approximation problem with non-symmetric weights: a theorem of Volberg is proved by elementary means.

math.CA↗

arXiv2020

On the number of zeros of functions in analytic quasianalytic classes

A space of analytic functions in the unit disc with uniformly continuous derivatives is said to be quasianalytic if the boundary value of a non-zero function from the class can not have a zero of infinite multiplicity. Such classes were described in the 1950-s and 1960-s by Carleson, Rodrigues-Salinas and Korenblum. A non-zero function from a quasianalytic space of analytic functions can only have a finite number of zeros in the closed disc. Recently, Borichev, Frank, and Volberg proved an explicit estimate on the number of zeros, for the case of quasianalytic Gevrey classes. Here, an estimate of similar form for general analytic quasianalytic classes is proved using a reduction to the classical quasianalyticity problem.

math.CA↗

arXiv2019

The density of states of 1D random band matrices via a supersymmetric transfer operator

Recently, T. and M. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator constructed from the supersymmetric integral representation for the density of states. We present a simpler proof of a slightly upgraded version of the semicircle law, which requires only standard semiclassical arguments and some peculiar elementary computations. The simplification is due to the use of supersymmetry, which manifests itself in the commutation between the transfer operator and a family of transformations of superspace, and was applied earlier in the context of band matrices by Constantinescu. Other versions of this supersymmetry have been a crucial ingredient in the study of the localization--delocalization transition by theoretical physicists.

math.PR↗

arXiv2018

Nonconventional Random Matrix Products

Let $ξ_1,ξ_2,...$ be independent identically distributed random variables and $F:\bbR^\ell\to SL_d(\bbR)$ be a Borel measurable matrix-valued function. Set $X_n=F(ξ_{q_1(n)},ξ_{q_2(n)},...,ξ_{q_\ell(n)})$ where $0\leq q_1<q_2<...<q_\ell$ are increasing functions taking on integer values on integers. We study the asymptotic behavior as $N\to\infty$ of the singular values of the random matrix product $Π_N=X_N\cdots X_2X_1$ and show, in particular, that (under certain conditions) $\frac 1N\log\|Π_N\|$ converges with probability one as $N\to\infty$. We also obtain similar results for such products when $ξ_i$ form a Markov chain. The essential difference from the usual setting appears since the sequence $(X_n)$ is long-range dependent and nonstationary.

math.PR↗

arXiv2018

The KPZ equation and moments of random matrices

The logarithm of the diagonal matrix element of a high power of a random matrix converges to the Cole-Hopf solution of the Kardar-Parisi-Zhang equation in the sense of one-point distributions.

math.PR↗