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Research area: Mathematics

Mathematics

Working on fundamental problems in mathematics and theoretical computer science, and interacting with the academic community and other researchers on challenging applied problems

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1,051 results found
Publication

A theoretical model for studying the nonlinear viscoelastic response of an active fluid undergoing oscillatory shear 

by Sara Malvar, Francisco Ricardo Cunha

In this work, a nonlinear phenomenological model for neutrally buoyant force-free active suspension of nematodes is proposed and tested. Just a few limited studies were found linked with nonlinear viscoelastic response of the active suspension investigated in this paper. The stress is decomposed through Fourier transform into elastic and viscous stress contributions. The stress response at large strain deviates drastically from the harmonic forcing in a nonlinear regime. In this case, the standard linear viscoelastic…

Publication

Acceleration via Fractal Learning Rate Schedules 

by Naman Agarwal, Surbhi Goel, Cyril Zhang

In practical applications of iterative first-order optimization, the learning rate schedule remains notoriously difficult to understand and expensive to tune. We demonstrate the presence of these subtleties even in the innocuous case when the objective is a convex quadratic. We reinterpret an iterative algorithm from the numerical analysis literature as what we call the Chebyshev learning rate schedule for accelerating vanilla gradient descent, and show that the problem of mitigating instability leads to a fractal…

Publication

Trellis BMA: coded trace reconstruction on IDS channels for DNA storage 

by Sundara Rajan Srinivasavaradhan; et. al.

Sequencing a DNA strand, as part of the read process in DNA storage, produces multiple noisy copies which can be combined to produce better estimates of the original strand; this is called trace reconstruction. One can reduce the error rate further by introducing redundancy in write sequence and this is called coded trace reconstruction. In this paper, we model the DNA storage channel as an insertion-deletion-substitution (IDS) channel and design both encoding schemes and low-complexity…

Publication

Batch optimization for DNA synthesis 

by Konstantin Makarychev; et. al.

Large pools of synthetic DNA molecules have been recently used to reliably store significant volumes of digital data. While DNA as a storage medium has enormous potential because of its high storage density, its practical use is currently severely limited because of the high cost and low throughput of available DNA synthesis technologies. We study the role of batch optimization in reducing the cost of large scale DNA synthesis, which translates to the following algorithmic…

Publication

Lower Bounds for Maximally Recoverable Tensor Code and Higher Order MDS Codes 

by Joshua Brakensiek, Sivakanth Gopi, Visu Makam

An $(m,n,a,b)$-tensor code consists of $m\times n$ matrices whose columns satisfy `$a$’ parity checks and rows satisfy `$b$’ parity checks (i.e., a tensor code is the tensor product of a column code and row code). Tensor codes are useful in distributed storage because a single erasure can be corrected quickly either by reading its row or column. Maximally Recoverable (MR) Tensor Codes, introduced by Gopalan et al.~\cite{GHKSWY}, are tensor codes which can correct every erasure…

A chart showing a GAN comparing fake images with real images, filtering them through a discriminator to produce a value indicating how fake the image is.
Microsoft Research Blog

How can generative adversarial networks learn real-life distributions easily 

A Generative adversarial network, or GAN, is one of the most powerful machine learning models proposed by Goodfellow et al. for learning to generate samples from complicated real-world distributions. GANs have sparked millions of applications, ranging from generating realistic images or cartoon characters to text-to-image translations. Turing award laureate Yann LeCun called GANs “the most interesting idea in the last 10 years in ML.” In the context of generating images, GANs consist of two parts.…

Publication

Numerical Composition of Differential Privacy 

by Sivakanth Gopi, Yin Tat Lee, Lukas Wutschitz

We give a fast algorithm to optimally compose privacy guarantees of differentially private (DP) algorithms to arbitrary accuracy. Our method is based on the notion of privacy loss random variables to quantify the privacy loss of DP algorithms. The running time and memory needed for our algorithm to approximate the privacy curve of a DP algorithm composed with itself k times is O(k^{1/2}). This improves over the best prior method by Koskela et al. (2020) which requires Ω(k^{1.5}) running time.…

Publication

Forward Super-Resolution: How Can GANs Learn Hierarchical Generative Models for Real-World Distributions 

by Zeyuan Allen-Zhu, Yuanzhi Li

Generative adversarial networks (GANs) are among the most successful models for learning high-complexity, real-world distributions. However, in theory, due to the highly non-convex, non-concave landscape of the minmax training objective, GAN remains one of the least understood deep learning models. In this work, we formally study how GANs can efficiently learn certain hierarchically generated distributions that are close to the distribution of images in practice. We prove that when a distribution has a structure that…

Dataset

Clustered Nanopore Reads (CNR) Dataset 

by Sivakanth Gopi, Sergey Yekhanin

DNA storage aims to store information in the form of DNA sequences. This is a research project in Microsoft Research Redmond. This repo contains a dataset of real DNA sequences which can be used for benchmarking different trace reconstruction algorithms. There is no code. We release the dataset of clustered nanopore DNA reads together with our paper: Trellis BMA: coded trace reconstruction on IDS channels for DNA storage

Group

Applied Research Group in Security 

We are an applied research team bringing impact driven research in the Security and Compliance suite for O365 customers. We explore cutting edge techniques in diverse disciplines that protect our customers from online attacks at scale, demonstrate Microsoft’s thought leadership in the security space, while also advancing state of the art in the process. Our research drives longer term product innovations and helps shape new product features and experiences at scale.  We actively cultivate internal…

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