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This paper introduces a new parameterization of deep neural networks (both fully-connected and convolutional) with guaranteed $\ell^2$ Lipschitz bounds, i.e. limited sensitivity to input perturbations. The Lipschitz guarantees are…

In $\mathbb{R}^n$, we establish an asymptotically sharp upper bound for the upper Minkowski dimension of $k$-porous sets having holes of certain size near every point in $k$ orthogonal directions at all small scales. This bound tends to…

The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Sepp\"{a}l\"ainen and Bal\'azs recently proved this conjecture for the nearest neighbor case. We…

Many algorithms have been recently proposed for causal machine learning. Yet, there is little to no theory on their quality, especially considering finite samples. In this work, we propose a theory based on generalization bounds that…

We count cycles of an unbounded length in generalized Johnson graphs. Asymptotics of the number of such cycles is obtained for certain growth rates of the cycle length.

This paper proposes Quantizable DNNs, a special type of DNNs that can flexibly quantize its bit-width (denoted as `bit modes' thereafter) during execution without further re-training. To simultaneously optimize for all bit modes, a…

Zero correlation zone (ZCZ) sequences and Golay complementary sequences are two kinds of sequences with different preferable correlation properties. Golay-ZCZ sequences are special kinds of complementary sequences which also possess a large…

We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to…

Numerical evidence suggests that certain permutation patterns of length k are easier to avoid than any other patterns of that same length. We prove that these patterns are avoided by no more than (2.25k^2)^n permutations of length n. In…

Deep neural network (NN) with millions or billions of parameters can perform really well on unseen data, after being trained from a finite training set. Various prior theories have been developed to explain such excellent ability of NNs,…

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.

We study the distribution of the length of longest increasing subsequences in random permutations of $n$ integers as $n$ grows large and establish an asymptotic expansion in powers of $n^{-1/3}$. Whilst the limit law was already shown by…

In this paper we mainly discuss sharp lower and upper bounds for the length of longest consecutive switches in IID Bernoulli sequences. This work is an extension of results in Erd\H{o}s and R\'{e}v\'{e}sz (1975) for longest head-run and Hao…

Let $(s_2(n))_{n=0}^\infty$ denote Stern's diatomic sequence. For $n\geq 2$, we may view $s_2(n)$ as the number of partitions of $n-1$ into powers of $2$ with each part occurring at most twice. More generally, for integers $b,n\geq 2$, let…

This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given n, sum the previous two terms and divide them by the largest possible power of n. The behavior of such sequences depends on n. We analyze…

An upper bound of composition series of groups of finite order is obtained. The bound is a nontrivial bound and so far best possible.

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…

Aimed at explaining the surprisingly good generalization behavior of overparameterized deep networks, recent works have developed a variety of generalization bounds for deep learning, all based on the fundamental learning-theoretic…

This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…

We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations…