Arithmetic Circuits, Structured Matrices and (not so) Deep Learning

Atri Rudra

Home/Archive/arXiv

arXiv · 2206.12490

Arithmetic Circuits, Structured Matrices and (not so) Deep Learning

Atri Rudra

View original source ↗Read source document ↗Export metadata ↓

Abstract

This survey presents a necessarily incomplete (and biased) overview of results at the intersection of arithmetic circuit complexity, structured matrices and deep learning. Recently there has been some research activity in replacing unstructured weight matrices in neural networks by structured ones (with the aim of reducing the size of the corresponding deep learning models). Most of this work has been experimental and in this survey, we formalize the research question and show how a recent work that combines arithmetic circuit complexity, structured matrices and deep learning essentially answers this question. This survey is targeted at complexity theorists who might enjoy reading about how tools developed in arithmetic circuit complexity helped design (to the best of our knowledge) a new family of structured matrices, which in turn seem well-suited for applications in deep learning. However, we hope that folks primarily interested in deep learning would also appreciate the connections to complexity theory.

Explore related subjects

cs.CC cs.LG

Keep this discovery

BibTeX RIS

Atri Rudra. 2022-06-24. Arithmetic Circuits, Structured Matrices and (not so) Deep Learning. https://arxiv.org/abs/2206.12490

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

cs.CC↗

arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗