Related papers: Weights at the Bottom Matter When the Top is Heavy

Let $d \ge 2, h \ge 1$ be integers. Using a fragmentation technique, we characterise $(h+1)$-tuples $(R_1, \dots, R_h, R)$ of non-empty families of partitions of $\{1, \dots, d\}$ such that it suffices for an order-$d$ tensor to have…

The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound. We prove that this bound can be sharp. We compute the…

We establish an explicit link between depth-3 formulas and one-sided approximation by depth-2 formulas, which were previously studied independently. Specifically, we show that the minimum size of depth-3 formulas is (up to a factor of n)…

Depth separation -- why a deeper network is more powerful than a shallower one -- has been a major problem in deep learning theory. Previous results often focus on representation power. For example, arXiv:1904.06984 constructed a function…

In this paper, we study the parameterized complexity of the MaxMin versions of two fundamental separation problems: Maximum Minimal $st$-Separator and Maximum Minimal Odd Cycle Transversal (OCT), both parameterized by the solution size. In…

Many multiagent tasks -- such as reviewer assignment, coalition formation, or fair resource allocation -- require selecting a group of agents such that collaboration remains effective even in the worst case. The \emph{weighted max-min…

In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of showing lower bound results. We define some natural restrictions of these models for which we are able to…

We establish new correlation bounds and pseudorandom generators for a collection of computation models. These models are all natural generalizations of structured low-degree $F_2$-polynomials that we did not have correlation bounds for…

Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}\in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a…

Let $d \ge 2$ be a positive integer. We show that for a class of notions $R$ of rank for order-$d$ tensors, which includes in particular the tensor rank, the slice rank and the partition rank, there exist functions $F_{d,R}$ and $G_{d,R}$…

In a recent breakthrough, Chen, Hirahara and Ren prove that $\mathsf{S_2E}/_1 \not\subset \mathsf{SIZE}[2^n/n]$ by giving a single-valued $\mathsf{FS_2P}$ algorithm for the Range Avoidance Problem ($\mathsf{Avoid}$) that works for…

Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an…

We refine several results of Bhatt-Morrow-Scholze on THH to THR. In particular, we compute THR of perfectoid rings. This will be useful for establishing motivic filtrations on real topological Hochschild and cyclic homology of quasisyntomic…

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03] established that…

Itsykson and Sokolov [IS14] identified resolution over parities, denoted by $\text{Res}(\oplus)$, as a natural and simple fragment of $\text{AC}^0[2]$-Frege for which no super-polynomial lower bounds on size of proofs are known. Building on…

A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…

We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical…

Tensor networks have been an important concept and technique in many research areas, such as quantum computation and machine learning. We study the exponential complexity of contracting tensor networks on two special graph structures:…

In the first part of this paper we investigate the operator aspect of higher-rank supersymmetric model which is introduced as a Lie theoretic extension of the $N=2$ minimal model with the simplest case $su(2)$ corresponding to the $N=2$…

We present a top-down lower-bound method for depth-$4$ boolean circuits. In particular, we give a new proof of the well-known result that the parity function requires depth-$4$ circuits of size exponential in $n^{1/3}$. Our proof is an…