arXiv subjects

Mihai Nica

Publications and source records attributed to Mihai Nica.

30 records · Page 2Linked to original sources

arXiv2021

One-sided reflected Brownian motions and the KPZ fixed point

We consider the system of one-sided reflected Brownian motions which is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov process defined in [MQR17] and believed to govern the long time large scale fluctuations for all models in the KPZ universality class. Brownian last passage percolation was shown recently in [DOV18] to converge to the Airy sheet (or directed landscape), defined there as a strong limit of a functional of the Airy line ensemble. This establishes the variational formula for the KPZ fixed point in terms of the Airy sheet.

math.PR↗

arXiv2021

RSK in last passage percolation: a unified approach

We present a version of the RSK correspondence based on the Pitman transform and geometric considerations. This version unifies ordinary RSK, dual RSK and continuous RSK. We show that this version is both a bijection and an isometry, two crucial properties for taking limits of last passage percolation models. We use the bijective property to give a non-computational proof that dual RSK maps Bernoulli walks to nonintersecting Bernoulli walks.

math.PR↗

arXiv2021

Intermediate disorder limits for multi-layer semi-discrete directed polymers

We show that the partition function of the multi-layer semi-discrete directed polymer converges in the intermediate disorder regime to the partition function for the multi-layer continuum polymer introduced by O'Connell and Warren. This verifies, modulo a previously hidden constant, an outstanding conjecture proposed by Corwin and Hammond. A consequence is the identification of the KPZ line ensemble as logarithms of ratios of consecutive layers of the continuum partition function. Other properties of the continuum partition function, such as continuity, strict positivity and contour integral formulas to compute mixed moments, are also identified from this convergence result.

math.PR↗

arXiv2020

Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning

The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for solutions based on samples of these Markov processes which have advantages over traditional numerical methods in some cases. However, naïve numerical implementations suffer from statistical bias and sampling error. We present methods to discretize the stochastic process appearing in the Feynman-Kac formula that reduce the bias of the numerical scheme. We also propose using temporal difference learning to assemble information from random samples in a way that is more efficient than the traditional Monte Carlo method.

math.NA↗

arXiv2020

A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks

We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic representation of the PDE in the spirit of the Feynman-Kac formula. The solution is given by an expectation of a martingale process driven by a Brownian motion. As Brownian walkers explore the domain, the deep neural network is iteratively trained using a form of reinforcement learning. Our method is a 'Derivative-Free Loss Method' since it does not require the explicit calculation of the derivatives of the neural network with respect to the input neurons in order to compute the training loss. The advantages of our method are showcased in a series of test problems: a corner singularity problem, an interface problem, and an application to a chemotaxis population model.

cs.LG↗

arXiv2020

Solution of the Kolmogorov equation for TASEP

We provide a direct and elementary proof that the formula obtained in [MQR17] for the TASEP transition probabilities for general (one-sided) initial data solves the Kolmogorov backward equation. The same method yields the solution for the related PushASEP particle system.

math.PR↗

arXiv2019

Finite Depth and Width Corrections to the Neural Tangent Kernel

We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not deterministic if depth and width simultaneously tend to infinity. Moreover, we prove that for such deep and wide networks, the NTK has a non-trivial evolution during training by showing that the mean of its first SGD update is also exponential in the ratio of network depth to width. This is sharp contrast to the regime where depth is fixed and network width is very large. Our results suggest that, unlike relatively shallow and wide networks, deep and wide ReLU networks are capable of learning data-dependent features even in the so-called lazy training regime.

cs.LG↗

arXiv2018

Products of Many Large Random Matrices and Gradients in Deep Neural Networks

We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $\ell_2$ norm of such a product applied to any fixed vector is asymptotically Gaussian. The fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity. Depending on the scaling limit considered, the mean and variance of the limiting Gaussian depend only on either the first two or the first four moments of the measure from which matrix entries are drawn. We also obtain explicit error bounds on the moments of the norm and the Kolmogorov-Smirnov distance to a Gaussian. Finally, we apply our result to obtain precise information about the stability of gradients in randomly initialized deep neural networks with ReLU activations. This provides a quantitative measure of the extent to which the exploding and vanishing gradient problem occurs in a fully connected neural network with ReLU activations and a given architecture.

math.PR↗

arXiv2018

The landscape of the spiked tensor model

We consider the problem of estimating a large rank-one tensor ${\boldsymbol u}^{\otimes k}\in({\mathbb R}^{n})^{\otimes k}$, $k\ge 3$ in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio $λ_{Bayes}= O(1)$ above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably no polynomial-time algorithm is known that achieved this goal unless $λ\ge C n^{(k-2)/4}$ and even powerful semidefinite programming relaxations appear to fail for $1\ll λ\ll n^{(k-2)/4}$. In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-$k$ homogeneous polynomial over the unit sphere in $n$ dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions $n$, and give exact formulas for the exponential growth rate. We show that (for $λ$ larger than a constant) critical points are either very close to the unknown vector ${\boldsymbol u}$, or are confined in a band of width $Θ(λ^{-1/(k-1)})$ around the maximum circle that is orthogonal to ${\boldsymbol u}$. For local maxima, this band shrinks to be of size $Θ(λ^{-1/(k-2)})$. These `uninformative' local maxima are likely to cause the failure of optimization algorithms.

math.ST↗

arXiv2017

Decorated Young Tableaux and the Poissonized Robinson-Schensted Process

We introduce an object called a decorated Young tableau which can equivalently be viewed as a continuous time trajectory of Young diagrams or as a non-intersecting line ensemble. By a natural extension of the Robinson-Schensted correspondence, we create a random pair of decorated Young tableaux from a Poisson point process in the plane, which we think of as a stochastic process in discrete space and continuous time. By using only elementary techniques and combinatorial properties, we identify this process as a Schur process and show it has the same law as certain non-intersecting Poisson walkers.

math.PR↗

arXiv2017

Intermediate disorder directed polymers and the multi-layer extension of the stochastic heat equation

We consider directed polymer models involving multiple non-intersecting random walks moving through a space-time disordered environment in one spatial dimension. For a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling (in which time and space are scaled diffusively, and the strength of the environment is scaled to zero in a critical manner) the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this paper we prove the analogous result for multiple non-intersecting random walks started and ended grouped together. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren.

math.PR↗

arXiv2013

Stabilization Time for a Type of Evolution on Binary Strings

We consider a type of evolution on {0,1}^n which occurs in discrete steps whereby at each step, we replace every occurrence of the substring "01" by "10". After at most n-1 steps we will reach a string of the form 11..1100..11, which we will call a "stabilized" string and we call the number of steps required the "stabilization time". If we choose each bit of the string independently to be a 1 with probability p and a 0 with probability 1-p, then the stabilization time of a string in {0,1}^n is a random variable with values in 0,1,...,n-1}. We study the asymptotic behavior of this random variable as n goes to infinity and we determine its limit distribution after suitable centering and scaling . When p is not 1/2, the limit distribution is Gaussian. When p = 1/2, the limit distribution is a χ_3 distribution. We also explicitly compute the limit distribution in a threshold setting where p=p_n varies with n given by p_n = 1/2 + λ/ 2 \sqrt{n} for λ> 0 a fixed parameter. This analysis gives rise to a one parameter family of distributions that fit between a χ_3 and a Gaussian distribution. The tools used in our arguments are a natural interpretation of strings in {0,1}^n as Young diagrams, and a connection with the known distribution for the maximal height of a Brownian path on [0,1].

math.PR↗