Michael Strand | Searcharxiv

arXiv2023

Block-missing data in linear systems: An unbiased stochastic gradient descent approach

Achieving accurate approximations to solutions of large linear systems is crucial, especially when those systems utilize real-world data. A consequence of using real-world data is that there will inevitably be missingness. Current approaches for dealing with missing data, such as deletion and imputation, can introduce bias. Recent studies proposed an adaptation of stochastic gradient descent (SGD) in specific missing-data models. In this work, we propose a new algorithm, $\ell$-tuple mSGD, for the setting in which data is missing in a block-wise, tuple pattern. We prove that our proposed method uses unbiased estimates of the gradient of the least squares objective in the presence of tuple missing data. We also draw connections between $\ell$-tuple mSGD and previously established SGD-type methods for missing data. Furthermore, we prove our algorithm converges when using updating step sizes and empirically demonstrate the convergence of $\ell$-tuple mSGD on synthetic data. Lastly, we evaluate $\ell$-tuple mSGD applied to real-world continuous glucose monitoring (CGM) device data.

math.NA↗

arXiv2022

Polynomial equations for matrices over integers modulo a prime power and the cokernel of a random matrix

Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in \mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ to the polynomial equation $P(X) = 0$, where $P(t)$ is a monic polynomial in $(\mathbb{Z}/p^{k}\mathbb{Z})[t]$ whose reduction modulo $p$ is square-free over the finite field $\mathbb{F}_{p}$ of $p$ elements. Noting that $P(X) = 0$ if and only if $\mathrm{cok}(P(X)) \simeq (\mathbb{Z}/p^{k}\mathbb{Z})^{n}$, we give a conjectural generalization of counting solutions to $P(X) = 0$ as the distribution of the cokernel $\mathrm{cok}(P(X))$ of $P(X)$ up to isomorphisms, where $X$ is a uniform random matrix in $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$. This distribution involves an explicit formula when we fix the residue class of $X$ modulo $p$. We prove this conjecture for the special case when the image of $P(t)$ in $\mathbb{F}_{p}[t]$ modulo $p$ is irreducible. We explain how the distribution we obtain is closely related to the Cohen-Lenstra distribution. Our proof involves algebraic and combinatorial arguments in linear algebra over $\mathbb{Z}/p^{k}\mathbb{Z}$ and builds upon a previous work of Cheong and Kaplan.

math.CO↗