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Mikhail Isaev

Publications and source records attributed to Mikhail Isaev.

At least 37 records · Page 2Linked to original sources
arXiv2022

Degree sequences of sufficiently dense random uniform hypergraphs

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random $r$-uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum.

math.CO↗
arXiv2022

Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions

We implement numerically formulas of [Isaev, Novikov, arXiv:2107.07882] for finding a compactly supported function $v$ on $\mathbb{R}^d$, $d\geq 1$, from its Fourier transform $\mathcal{F} [v]$ given within the ball $B_r$. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for $d = 1$. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit.

math.NA↗
arXiv2021

Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions

We give new formulas for finding a compactly supported function $v$ on $\mathbb{R}^d$, $d\geq 1$, from its Fourier transform $\mathcal{F} v$ given within the ball $B_r$. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWFs). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given.

math.CA↗
arXiv2021

Spanning trees in random regular uniform hypergraphs

Let $\mathcal{G}_{n,r,s}$ denote a uniformly random $r$-regular $s$-uniform hypergraph on the vertex set $\{1,2,\ldots, n\}$. We establish a threshold result for the existence of a spanning tree in $\mathcal{G}_{n,r,s}$, restricting to $n$ satisfying the necessary divisibility conditions. Specifically, we show that when $s\geq 5$, there is a positive constant $ρ(s)$ such that for any $r\geq 2$, the probability that $\mathcal{G}_{n,r,s}$ contains a spanning tree tends to 1 if $r > ρ(s)$, and otherwise this probability tends to zero. The threshold value $ρ(s)$ grows exponentially with $s$. As $\mathcal{G}_{n,r,s}$ is connected with probability which tends to 1, this implies that when $r \leq ρ(s)$, most $r$-regular $s$-uniform hypergraphs are connected but have no spanning tree. When $s=3,4$ we prove that $\mathcal{G}_{n,r,s}$ contains a spanning tree with probability which tends to 1, for any $r\geq 2$. Our proof also provides the asymptotic distribution of the number of spanning trees in $\mathcal{G}_{n,r,s}$ for all fixed integers $r,s\geq 2$. TPreviously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

math.CO↗
arXiv2020

Subgraph counts for dense random graphs with specified degrees

We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence $(d_1,\ldots,d_n)$ as $n \rightarrow \infty$. We also determine the expected number of spanning trees in this model. The range of degrees covered includes $d_j = λn + O(n^{1/2+\varepsilon})$ for some $λ$ bounded away from $0$ and $1$.

math.CO↗
arXiv2020

Hölder-logarithmic stability in Fourier synthesis

We prove a Hölder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function $v$ on $\mathbb{R}^d$ from its Fourier transform $\mathcal{F} v$ given on $[-r,r]^d$. This estimate relies on a Hölder stable continuation of $\mathcal{F}v$ from $[-r,r]^d$ to a larger domain. The related reconstruction procedures are based on truncated series of Chebyshev polynomials. We also give an explicit example showing optimality of our stability estimates.

math.CA↗
arXiv2020

Stability estimates for reconstruction from the Fourier transform on the ball

We prove Hölder-logarithmic stability estimates for the problem of finding an integrable function $v$ on $\mathbb{R}^d$ with a super-exponential decay at infinity from its Fourier transform $\mathcal{F} v$ given on the ball $B_r$. These estimates arise from a Hölder-stable extrapolation of $\mathcal{F} v$ from $B_r$ to a larger ball. We also present instability examples showing an optimality of our results.

math.CA↗
arXiv2020

Integer Quantization for Deep Learning Inference: Principles and Empirical Evaluation

Quantization techniques can reduce the size of Deep Neural Networks and improve inference latency and throughput by taking advantage of high throughput integer instructions. In this paper we review the mathematical aspects of quantization parameters and evaluate their choices on a wide range of neural network models for different application domains, including vision, speech, and language. We focus on quantization techniques that are amenable to acceleration by processors with high-throughput integer math pipelines. We also present a workflow for 8-bit quantization that is able to maintain accuracy within 1% of the floating-point baseline on all networks studied, including models that are more difficult to quantize, such as MobileNets and BERT-large.

cs.LG↗
arXiv2020

Asymptotic enumeration of orientations of a graph as a function of the out-degree sequence

We prove an asymptotic formula for the number of orientations with given out-degree (score) sequence for a graph $G$. The graph $G$ is assumed to have average degrees at least $n^{1/3 + \varepsilon}$ for some $\varepsilon > 0$, and to have strong mixing properties, while the maximum imbalance (out-degree minus in-degree) of the orientation should be not too large. Our enumeration results have applications to the study of subdigraph occurrences in random orientations with given imbalance sequence. As one step of our calculation, we obtain new bounds for the maximum likelihood estimators for the Bradley-Terry model of paired comparisons.

math.CO↗
arXiv2019

A threshold result for loose Hamiltonicity in random regular uniform hypergraphs

Let $\mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniform hypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may grow with $n$. An $\ell$-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely $\ell$ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When $r,s\geq 3$ are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Rucinski and Sileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in $\mathcal{G}(n,r,s)$. Finally we prove that for $\ell = 2,\ldots, s-1$ and for $r$ growing moderately as $n\to\infty$, the probability that $\mathcal{G}(n,r,s)$ has a $\ell$-overlapping Hamilton cycle tends to zero.

math.CO↗
arXiv2017

Complex martingales and asymptotic enumeration

Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic behaviour of such integrals, we establish explicit bounds on the exponentials of complex martingales. Those bounds applied to the case of truncated normal distributions are precise enough to include and extend many enumerative results of Barvinok, Canfield, Gao, Greenhill, Hartigan, Isaev, McKay, Wang, Wormald, and others. Our method applies to sums as well as integrals. As a first illustration of the power of our theory, we considerably strengthen existing results on the relationship between random graphs or bipartite graphs with specified degrees and the so-called $β$-model of random graphs with independent edges, which is equivalent to the Rasch model in the bipartite case.

math.CO↗
arXiv2017

The average number of spanning trees in sparse graphs with given degrees

We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence $\boldsymbol{d}=(d_1,\ldots, d_n)$, provided that the number of edges is at least $n + \textstyle{\frac{1}{2}} d_{\max}^4$, where $d_{\max}$ is the maximum degree. A key part of our argument involves establishing a concentration result for a certain family of functions over random trees with given degrees, using Prüfer codes.

math.CO↗
arXiv2016

On a bound of Hoeffding in the complex case

It was proved by Hoeffding in 1963 that a real random variable X confined to [a, b] satisfies E e^(X--E X) $\le$ e^((b--a)^2/8). We generalise this to complex random variables.

math.PR↗
arXiv2014

Effectivized Holder-logarithmic stability estimates for the Gel'fand inverse problem

We give effectivized Holder-logarithmic energy and regularity dependent stability estimates for the Gel'fand inverse boundary value problem in dimension $d=3$. This effectivization includes explicit dependance of the estimates on coefficient norms and related parameters. Our new estimates are given in $L^2$ and $L^\infty$ norms for the coefficient difference and related stability efficiently increases with increasing energy and/or coefficient difference regularity. Comparisons with preceeding results are given.

math.AP↗
arXiv2013

On the class of graphs with strong mixing properties

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties. In addition, we present asymptotic formulas for the numbers of Eulerian orientations and Eulerian circuits in an undirected simple graph.

math.CO↗
arXiv2012

Reconstruction of a potential from the impedance boundary map

We give formulas and equations for finding generalized scattering data for the Schrödinger equation in open bounded domain at fixed energy from the impedance boundary map (or Robin-to-Robin map). Combining these results with results of the inverse scattering theory we obtain efficient methods for reconstructing potential from the impedance boundary map.

math.AP↗
arXiv2012

Energy and regularity dependent stability estimates for near-field inverse scattering in multidimensions

We prove new global Hölder-logarithmic stability estimates for the near-field inverse scattering problem in dimension $d\geq 3$. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. In addition, a global logarithmic stability estimate for this inverse problem in dimension $d=2$ is also given.

math.AP↗