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Researcher profile

Dan Mikulincer

Dan Mikulincer contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.PRMachine Learningmath.FAmath.STStatistics TheoryInformation Theorymath.ITSocial and Information NetworksCryptography and SecurityData Structures and Algorithmsmath.CAmath.MGmath.OC

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Published work

10 published item(s)

preprint2026arXiv

The Benefits of Temporal Correlations: SGD Learns k-Juntas from Random Walks Efficiently

We study how temporal correlations in the data can make certain sparse learning problems efficiently learnable by gradient-based methods. Our focus is on Boolean k-juntas, a canonical sparse learning problem known to pose barriers for gradient-based methods under independent uniform samples. We show that this picture changes when the samples are generated by a lazy random walk on the hypercube. In this setting, the temporal dependencies can be exploited by a two-layer ReLU network trained using stylized-SGD with a temporal-difference loss, which compares target and predicted increments across consecutive samples. For every fixed k, the resulting sample complexity is essentially linear in the ambient dimension d. By contrast, we show that for large-batch gradient methods using standard convex pointwise losses, temporal correlations do not provide the same advantage.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Anti-concentration of polynomials: dimension-free covariance bounds and decay of Fourier coefficients

We study random variables of the form $f(X)$, when $f$ is a degree $d$ polynomial, and $X$ is a random vector on $\mathbb{R}^{n}$, motivated towards a deeper understanding of the covariance structure of $X^{\otimes d}$. For applications, the main interest is to bound $\mathrm{Var}(f(X))$ from below, assuming a suitable normalization on the coefficients of $f$. Our first result applies when $X$ has independent coordinates, and we establish dimension-free bounds. We also show that the assumption of independence can be relaxed and that our bounds carry over to uniform measures on isotropic $L_{p}$ balls. Moreover, in the case of the Euclidean ball, we provide an orthogonal decomposition of $\mathrm{Cov}(X^{\otimes d})$. Finally, we utilize the connection between anti-concentration and decay of Fourier coefficients to prove a high-dimensional analogue of the van der Corput lemma, thus partially answering a question posed by Carbery and Wright.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Archimedes Meets Privacy: On Privately Estimating Quantiles in High Dimensions Under Minimal Assumptions

The last few years have seen a surge of work on high dimensional statistics under privacy constraints, mostly following two main lines of work: the ``worst case'' line, which does not make any distributional assumptions on the input data; and the ``strong assumptions'' line, which assumes that the data is generated from specific families, e.g., subgaussian distributions. In this work we take a middle ground, obtaining new differentially private algorithms with polynomial sample complexity for estimating quantiles in high-dimensions, as well as estimating and sampling points of high Tukey depth, all working under very mild distributional assumptions. From the technical perspective, our work relies upon deep robustness results in the convex geometry literature, demonstrating how such results can be used in a private context. Our main object of interest is the (convex) floating body (FB), a notion going back to Archimedes, which is a robust and well studied high-dimensional analogue of the interquantile range. We show how one can privately, and with polynomially many samples, (a) output an approximate interior point of the FB -- e.g., ``a typical user'' in a high-dimensional database -- by leveraging the robustness of the Steiner point of the FB; and at the expense of polynomially many more samples, (b) produce an approximate uniform sample from the FB, by constructing a private noisy projection oracle.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Community detection and percolation of information in a geometric setting

We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Noise stability on the Boolean hypercube via a renormalized Brownian motion

We consider a variant of the classical notion of noise on the Boolean hypercube which gives rise to a new approach to inequalities regarding noise stability. We use this approach to give a new proof of the Majority is Stablest theorem by Mossel, O'Donnell, and Oleszkiewicz, improving the dependence of the bound on the maximal influence of the function from logarithmic to polynomial. We also show that a variant of the conjecture by Courtade and Kumar regarding the most informative Boolean function, where the classical noise is replaced by our notion, holds true. Our approach is based on a stochastic construction that we call the renormalized Brownian motion, which facilitates the use of inequalities in Gaussian space in the analysis of Boolean functions.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Non-asymptotic approximations of neural networks by Gaussian processes

We study the extent to which wide neural networks may be approximated by Gaussian processes when initialized with random weights. It is a well-established fact that as the width of a network goes to infinity, its law converges to that of a Gaussian process. We make this quantitative by establishing explicit convergence rates for the central limit theorem in an infinite-dimensional functional space, metrized with a natural transportation distance. We identify two regimes of interest; when the activation function is polynomial, its degree determines the rate of convergence, while for non-polynomial activations, the rate is governed by the smoothness of the function.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

How to trap a gradient flow

We consider the problem of finding an $\varepsilon$-approximate stationary point of a smooth function on a compact domain of $\mathbb{R}^d$. In contrast with dimension-free approaches such as gradient descent, we focus here on the case where $d$ is finite, and potentially small. This viewpoint was explored in 1993 by Vavasis, who proposed an algorithm which, for any fixed finite dimension $d$, improves upon the $O(1/\varepsilon^2)$ oracle complexity of gradient descent. For example for $d=2$, Vavasis' approach obtains the complexity $O(1/\varepsilon)$. Moreover for $d=2$ he also proved a lower bound of $Ω(1/\sqrt{\varepsilon})$ for deterministic algorithms (we extend this result to randomized algorithms). Our main contribution is an algorithm, which we call gradient flow trapping (GFT), and the analysis of its oracle complexity. In dimension $d=2$, GFT closes the gap with Vavasis' lower bound (up to a logarithmic factor), as we show that it has complexity $O\left(\sqrt{\frac{\log(1/\varepsilon)}{\varepsilon}}\right)$. In dimension $d=3$, we show a complexity of $O\left(\frac{\log(1/\varepsilon)}{\varepsilon}\right)$, improving upon Vavasis' $O\left(1 / \varepsilon^{1.2} \right)$. In higher dimensions, GFT has the remarkable property of being a logarithmic parallel depth strategy, in stark contrast with the polynomial depth of gradient descent or Vavasis' algorithm. In this higher dimensional regime, the total work of GFT improves quadratically upon the only other known polylogarithmic depth strategy for this problem, namely naive grid search. We augment this result with another algorithm, named \emph{cut and flow} (CF), which improves upon Vavasis' algorithm in any fixed dimension.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Information and dimensionality of anisotropic random geometric graphs

This paper deals with the problem of detecting non-isotropic high-dimensional geometric structure in random graphs. Namely, we study a model of a random geometric graph in which vertices correspond to points generated randomly and independently from a non-isotropic $d$-dimensional Gaussian distribution, and two vertices are connected if the distance between them is smaller than some pre-specified threshold. We derive new notions of dimensionality which depend upon the eigenvalues of the covariance of the Gaussian distribution. If $α$ denotes the vector of eigenvalues, and $n$ is the number of vertices, then the quantities $\left(\frac{||α||_2}{||α||_3}\right)^6/n^3$ and $\left(\frac{||α||_2}{||α||_4}\right)^4/n^3$ determine upper and lower bounds for the possibility of detection. This generalizes a recent result by Bubeck, Ding, Rácz and the first named author from [BDER14] which shows that the quantity $d/n^3$ determines the boundary of detection for isotropic geometry. Our methods involve Fourier analysis and the theory of characteristic functions to investigate the underlying probabilities of the model. The proof of the lower bound uses information theoretic tools, based on the method presented in [BG15].

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

The CLT in high dimensions: quantitative bounds via martingale embedding

We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy, and in particular: (a) We improve the best known bound, obtained by the third named author, for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) We derive the first non-asymptotic convergence rate for the entropic CLT in arbitrary dimension, for general log-concave random vectors; (c) We give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity. Our method is based on martingale embeddings and specifically on the Skorokhod embedding constructed by the first named author.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Stability of the Shannon-Stam inequality via the Föllmer process

We prove stability estimates for the Shannon-Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors $X,Y \in \mathbb{R}^d$, the deficit in the Shannon-Stam inequality is bounded from below by the expression $$ C \left(\mathrm{D}\left(X||G\right) + \mathrm{D}\left(Y||G\right)\right), $$ where $\mathrm{D}\left( \cdot ~ ||G\right)$ denotes the relative entropy with respect to the standard Gaussian and the constant $C$ depends only on the covariance structures and the spectral gaps of $X$ and $Y$. In the case of uniformly log-concave vectors our analysis gives dimension-free bounds. Our proofs are based on a new approach which uses an entropy-minimizing process from stochastic control theory.

0 citations0 reviewsNo code linkedOpen access