arXiv subjects

Todd Kemp

Publications and source records attributed to Todd Kemp.

At least 19 recordsLinked to original sources

arXiv2026

Eigen-Spike Emergence and Quadratic Equivalents for Conjugate Kernels on Nonlinearly Separable Data

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in neural networks (NNs). Such equivalents make theoretical predictions tractable by reducing a complex model to a simpler one with properties that fall under the umbrella of classical RMT tools. However, this leaves open the question of whether this idealized linear equivalence remains meaningful for classification of high-dimensional nonlinearly separable data. Motivated by this, we consider the conjugate kernel (CK), which is the nonlinear feature map of a one-layer feedforward NN, under a canonical nonlinearly separable dataset for the XOR problem; and we use the study of informative outlier eigenvalues in the CK and whether their corresponding eigenvectors asymptotically align with XOR labels as a proxy for nonlinear learnability. We develop a robust quadratic equivalent of the CK matrix that enables a precise analysis of emergent informative spikes, as one modifies various knobs common in ML practice: sample complexity, signal-to-noise ratio (SNR), nonlinear activation choice, and pretrained features. We identify regimes in which these knobs move the CK beyond the linear equivalent and produce BBP-type transitions to label-aligned outlier eigenspaces. Our analysis helps bring deterministic-equivalence tools from RMT to bear on problems of practical relevance in ML.

stat.ML↗

arXiv2025

Double $q$-Wigner Chaos and the Fourth Moment

In this paper, we prove the Fourth Moment Theorem for sequences of (noncommutative) random variables given as sums of two stochastic integrals in two different parity orders of chaos, both in the free Wigner chaos setting and a $q$-Gaussian generalization. Specifically, we prove that convergence to the appropriate central limit distribution is mediated entirely by the behavior of the first four (mixed) moments of the two stochastic integrals, which in turn controls the $L^2$ norms of partial integral contractions of those kernels. The key step in both the free and $q$-Gaussian settings is a polarization identity for fourth cumulants of sums which holds only when the two terms have differing parities. These results are analogous to the recent preprint Fourth-Moment Theorems for Sums of Multiple Integrals by Basse-O'Connor, Kramer-Bang, and Svedsen in the classical Wiener-It\^o chaos setting.

math.PR↗

arXiv2025

Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$

We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group $\mathrm{GL}(N,\mathbb{C})$ converges almost surely to a deterministic probability measure, characterized by a free stochastic differential equation. This fully resolves a conjecture made by Philippe Biane in 1997. Our analysis includes a family $\{B=B_{\rho,\zeta}\colon |\zeta|<\rho\}$ of nondegenerate diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ whose laws are invariant under unitary conjugation, with initial distributions assumed to be uniformly bounded and invertible. The crux of our analysis is a strong quantitative approximation of Brownian motion $B(t)$ on $\mathrm{GL}(N,\mathbb{C})$ for small $t$ by a single increment $I+W(t)$, where $W=W_{\rho,\zeta}$ is an elliptic Brownian motion in the Lie algebra $\mathfrak{gl}(N,\mathbb{C}) = \mathbb{M}_N(\mathbb{C})$. Specifically, for any $t\in[0,1]$ and $\delta>0$, \[ \mathbb{P}\left(\|B(t)-I-W(t)\|\geq \delta\right)\leq \left(C t/\delta\right)^{N^{2/3}} \] for a constant $C=C_\rho$. Leveraging independence of multiplicative increments of the Brownian motion then allows us to use powerful (anti-)concentration tools for Gaussian matrices to complete the Hermitization procedure for convergence of eigenvalues.

math.PR↗

arXiv2025

Matrix Random Walks and the Lima Bean Law

A matrix random walk is a stochastic process of the form $B_k = (I+A_1)\cdots(I+A_k)$ where $A_j$ are independent ``step'' matrices in $\mathrm{M}_N(\mathbb{C})$. With the right entry-covariance, a rescaled matrix random walk converges to Brownian motion $B(t)$ on a matrix Lie group. In this paper, we study the eigenvalues of such rescaled matrix random walks, as $N\to\infty$ and $k\to\infty$. The standard Brownian motion $W(t)$ on $\mathrm{M}_N(\mathbb{C})$ has independent Gaussian entries at each $t$. It is bi-invariant: mutiplying on the left or right by a unitary does not change the distribution. We prove that the empirical eigenvalue distribution of any matrix random walk $B_k$ with bi-invariant steps $A_j$ and initial distribution converges (for fixed $k$ as $N\to\infty$) to a probability measure on $\mathbb{C}$: the Brown measure of the free probability $\ast$-distribution limit $b_k$ of the random walk. If the steps $A_j$ are identically distributed with normalized Hilbert--Schmidt norm $\|A_j\|_2 = t$, the limit law of eigenvalues is supported on a compact ``lima bean'' shaped region. We explicitly compute the limit measure and region, and characterize their phase transitions as $t$ evolves. We prove that the Brown measure of $b_k$ converges as $k\to\infty$, to the Brown measure of the free multiplicative Brownian motion, assuming only that the steps are bi-invariant and normalized in Hilbert--Schmidt norm. Thus the Brownian motion is the universal limit of rescaled matrix random walks, under very general assumptions on the distribution of steps.

math.PR↗

arXiv2024

The strong Haagerup inequality for q-circular systems

Together with Speicher, in 2007 the first author proved the strong Haagerup inequality for operator norms of homogeneous holomorphic polynomials in freely independent $\mathscr{R}$-diagonal elements (including in particular circular random variables); the inequality improved the bound from the original Haagerup inequality to grow with $\sqrt{n}$, rather than linearly in $n$, on homogeneous polynomials of degree $n$. In this paper, we prove a similar inequality for $q$-circular systems for $|q|<1$, generalizing the free case when $q=0$. In particular, we prove the strong Haagerup inequality for systems exhibiting neither free independence nor $\mathscr{R}$-diagonality. As an application, we prove a strong ultracontractivity theorem for the $q$-Ornstein--Uhlenbeck semigroup, and prove sharp rates for the Haagerup and ultracontractive inequalities.

math.OA↗

arXiv2024

Bias and Division in the Free World

Sampling bias is a foundational concept in statistics; associated bias transforms, such as size bias, have come to play important roles in probability theory of late. The first author and G. Reinert introduced zero bias, a transform whose unique fixed point is the normal distribution; it has become a standard tool in Stein's method and Gaussian approximation. Very recently, connections between zero bias and the class of infinitely divisible distributions have been found. In this paper, we develop a free probabilistic analog of the zero bias transform, proving its existence and regularity. The free zero bias has the semicircle law (free probability's central limit distribution) as its unique fixed point. We offer a construction of the free zero bias that mirrors a classical one incorporating square bias with a mollifier, and in the process develop a surprisingly new class of distributional operations through their Cauchy transforms. We then explore connections between the free zero bias, and size bias, with the class of freely infinitely divisible distributions. We develop a new self-contained treatment of the subject, together with a new characterization of free infinite divisibility using bias transforms. We also develop a parallel treatment of positively freely infinitely divisible distributions, which can also be characterized by a new kind of Levy--Khintchine formula that has no known classical analogue, and we use this to both give several new descriptions of such distributions and furnish new examples using these bias methods.

math.PR↗

arXiv2019

The Brown measure of the free multiplicative Brownian motion

The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $\mathsf{GL}(N;\mathbb{C}),$ in the sense of $\ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $\Sigma_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $\bar{\Sigma}_{t},$ which is strictly positive and real analytic on $\Sigma_{t}$. This density has a simple form in polar coordinates: \[ W_{t}(r,\theta)=\frac{1}{r^{2}}w_{t}(\theta), \] where $w_{t}$ is an analytic function determined by the geometry of the region $\Sigma_{t}$. We show also that the spectral measure of free unitary Brownian motion $u_{t}$ is a "shadow" of the Brown measure of $b_{t}$, precisely mirroring the relationship between Wigner's semicircle law and Ginibre's circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

math.PR↗

arXiv2018

Brown Measure Support and the Free Multiplicative Brownian Motion

The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of Brownian motion $B_t^N$ on the general linear group $\mathrm{GL}(N;\mathbb{C})$. We prove that the Brown measure for $b_{t}$---which is an analog of the empirical eigenvalue distribution for matrices---is supported on the closure of a certain domain $\Sigma_{t}$ in the plane. The domain $\Sigma_t$ was introduced by Biane in the context of the large-$N$ limit of the Segal--Bargmann transform associated to $\mathrm{GL}(N;\mathbb{C})$. We also consider a two-parameter version, $b_{s,t}$: the large-$N$ limit of a related family of diffusion processes on $\mathrm{GL}(N;\mathbb{C})$ introduced by the second author. We show that the Brown measure of $b_{s,t}$ is supported on the closure of a certain planar domain $\Sigma_{s,t}$, generalizing $\Sigma_t$, introduced by Ho. In the process, we introduce a new family of spectral domains related to any operator in a tracial von Neumann algebra: the {\em $L^p_n$-spectrum} for $n\in\mathbb{N}$ and $p\ge 1$, a subset of the ordinary spectrum defined relative to potentially-unbounded inverses. We show that, in general, the support of the Brown measure of an operator is contained in its $L_2^2$-spectrum.

math.FA↗

arXiv2016

Most Boson quantum states are almost maximally entangled

The geometric measure of entanglement $E$ of an $m$ qubit quantum state takes maximal possible value $m$. In previous work of Gross, Flammia, and Eisert, it was shown that $E \ge m-O(\log m)$ with high probability as $m\to\infty$. They showed, as a consequence, that the vast majority of states are too entangled to be computationally useful. In this paper, we show that for $m$ qubit {\em Boson} quantum states (those that are actually available in current designs for quantum computers), the maximal possible geometric measure of entanglement is $\log_2 m$, opening the door to many computationally universal states. We further show the corresponding concentration result that $E \ge \log_2 m - O(\log \log m)$ with high probability as $m\to\infty$. We extend these results also to $m$-mode $n$-bit Boson quantum states.

math-ph↗

arXiv2016

The Complex-Time Segal-Bargmann Transform

We introduce a new form of the Segal--Bargmann transform for a Lie group $K$ of compact type. We show that the heat kernel $(\rho_{t}(x))_{t>0,x\in K}$ has a space-time analytic continuation to a holomorphic function \[ (\rho_{\mathbb{C}}(\tau,z))_{\mathrm{Re}\,\tau>0,z\in K_{\mathbb{C}}} \] where $K_{\mathbb{C}}$ is the complexification of $K$. The new transform is defined by the integral \[ (B_{\tau}f)(z)=\int_{K}\rho_{\mathbb{C}}(\tau,zk^{-1})f(k)\,dk,\quad z\in K_{\mathbb{C}}. \] If $s>0$ and $\tau\in\mathbb{D}(s,s)$ (the disk of radius $s$ centered at $s$), this integral defines a holomorphic function on $K_{\mathbb{C}}$ for each $f\in L^{2}(K,\rho_{s})$. We construct a heat kernel density $\mu_{s,\tau}$ on $K_{\mathbb{C}}$ such that, for all $s,\tau$ as above, $B_{s,\tau}:=B_{\tau}|_{L^{2}(K,\rho_{s})}$ is an isometric isomorphism from $L^{2}(K,\rho_{s})$ onto the space of holomorphic functions in $L^{2}(K_{\mathbb{C}},\mu_{s,\tau})$. When $\tau=t=s$, the transform $B_{t,t}$ coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When $\tau=t\in (0,2s)$, the transform $B_{s,t}$ coincides with the one introduced by the first two authors.

math.FA↗

arXiv2016

The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces

We prove the Makeenko-Migdal equation for two-dimensional Euclidean Yang-Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.

math-ph↗

arXiv2016

Three proofs of the Makeenko-Migdal equation for Yang-Mills theory on the plane

We give three short proofs of the Makeenko-Migdal equation for the Yang-Mills measure on the plane, two using the edge variables and one using the loop or lasso variables. Our proofs are significantly simpler than the earlier pioneering rigorous proofs given by T. L\'evy and by A. Dahlqvist. In particular, our proofs are "local" in nature, in that they involve only derivatives with respect to variables adjacent to the crossing in question. In an accompanying paper with F. Gabriel, we will show that two of our proofs can be adapted to the case of Yang-Mills theory on a compact surface.

math-ph↗

arXiv2016

The minimum Renyi entropy output of a quantum channel is locally additive

We show that the minimum Renyi entropy output of a quantum channel is locally additive for Renyi parameter alpha>1. While our work extends the results of [10] (in which local additivity was proven for alpha=1), it is based on several new techniques that incorporate the multiplicative nature of p-norms, in contrast to the additivity property of the von-Neumann entropy. Our results demonstrate that the counterexamples to the Renyi additivity conjectures exhibit global effects of quantum channels. Interestingly, the approach presented here can not be extended to Renyi entropies with parameter alpha<1.

quant-ph↗

arXiv2015

Fluctuations of Brownian Motions on GL_N

We consider a two parameter family of unitarily invariant diffusion processes on the general linear group $\mathbb{GL}_N$ of $N\times N$ invertible matrices, that includes the standard Brownian motion as well as the usual unitary Brownian motion as special cases. We prove that all such processes have Gaussian fluctuations in high dimension with error of order $O(1/N)$; this is in terms of the finite dimensional distributions of the process under a large class of test functions known as trace polynomials. We give an explicit characterization of the covariance of the Gaussian fluctuation field, which can be described in terms of a fixed functional of three freely independent free multiplicative Brownian motions. These results generalize earlier work of Lévy and Maïda, and Diaconis and Evans, on unitary groups. Our approach is geometric, rather than combinatorial.

math.PR↗

arXiv2015

Strong Convergence of Unitary Brownian Motion

The Brownian motion $(U^N_t)_{t\ge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t)_{t\ge 0}$ as $N\to\infty$. In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time $t>0$, we prove that the spectral measure has a hard edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.

math.PR↗

arXiv2014

On Sharp Constants for Dual Segal--Bargmann $L^p$ Spaces

We study dilated holomorphic $L^p$ space of Gaussian measures over $\mathbb{C}^n$, denoted $\mathcal{H}_{p,α}^n$ with variance scaling parameter $α>0$. The duality relations $(\mathcal{H}_{p,α}^n)^\ast \cong \mathcal{H}_{p',α}$ hold with $\frac{1}{p}+\frac{1}{p'}=1$, but not isometrically. We identify the sharp lower constant comparing the norms on $\mathcal{H}_{p',α}$ and $(\mathcal{H}_{p,α}^n)^\ast$, and provide upper and lower bounds on the sharp upper constant. We prove several suggestive partial results on the sharpness of the upper constant. One of these partial results leads to a sharp bound on each Taylor coefficient of a function in the Fock space for $n=1$.

math.FA↗

arXiv2014

Random Matrices with Log-Range Correlations, and Log-Sobolev Inequalities

Let $X_N$ be a symmetric $N\times N$ random matrix whose $\sqrt{N}$-scaled centered entries are uniformly square integrable. We prove that if the entries of $X_N$ can be partitioned into independent subsets each of size $o(\log N)$, then the empirical eigenvalue distribution of $X_N$ converges weakly to its mean in probability. This significantly extends the best previously known results on convergence of eigenvalues for matrices with correlated entries (where the partition subsets are blocks and of size $O(1)$.) we prove this result be developing a new log-Sobolev inequality, generalizing the first author's introduction of mollified log-Sobolev inequalities: we show that if $\mathbf{Y}$ is a bounded random vector and $\mathbf{Z}$ is a standard normal random vector independent from $\mathbf{Y}$, then the law of $\mathbf{Y}+t\mathbf{Z}$ satisfies a log-Sobolev inequality for all $t>0$, and we give bounds on the optimal log-Sobolev constant.

math.FA↗

arXiv2013

Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through log-subharmonic functions, and use it to prove the equivalence of strong hypercontractivity and the strong log Sobolev inequality for such log-subharmonic functions.

math.FA↗