arXiv subjects

Shyam Narayanan

Publications and source records attributed to Shyam Narayanan.

47 records · Page 3Linked to original sources

arXiv2021

Learning-based Support Estimation in Sublinear Time

We consider the problem of estimating the number of distinct elements in a large data set (or, equivalently, the support size of the distribution induced by the data set) from a random sample of its elements. The problem occurs in many applications, including biology, genomics, computer systems and linguistics. A line of research spanning the last decade resulted in algorithms that estimate the support up to $ \pm \varepsilon n$ from a sample of size $O(\log^2(1/\varepsilon) \cdot n/\log n)$, where $n$ is the data set size. Unfortunately, this bound is known to be tight, limiting further improvements to the complexity of this problem. In this paper we consider estimation algorithms augmented with a machine-learning-based predictor that, given any element, returns an estimation of its frequency. We show that if the predictor is correct up to a constant approximation factor, then the sample complexity can be reduced significantly, to \[ \ \log (1/\varepsilon) \cdot n^{1-Θ(1/\log(1/\varepsilon))}. \] We evaluate the proposed algorithms on a collection of data sets, using the neural-network based estimators from {Hsu et al, ICLR'19} as predictors. Our experiments demonstrate substantial (up to 3x) improvements in the estimation accuracy compared to the state of the art algorithm.

cs.LG↗

arXiv2021

Arithmetic Progressions in Sumsets of Sparse Sets

A set of positive integers $A \subset \mathbb{Z}_{> 0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we study arithmetic progressions in sums of log-sparse subsets of $\mathbb{Z}_{> 0}$. We prove that for any log-sparse subsets $S_1, \dots, S_n$ of $\mathbb{Z}_{> 0},$ the sumset $S = S_1 + \cdots + S_n$ cannot contain an arithmetic progression of size greater than $n^{(1+o(1))n}.$ We also show that this is nearly tight by proving that there exist log-sparse sets $S_1, \dots, S_n$ such that $S_1 + \cdots + S_n$ contains an arithmetic progression of size $n^{(1-o(1)) n}.$

math.CO↗

arXiv2020

Circular Trace Reconstruction

Trace reconstruction is the problem of learning an unknown string $x$ from independent traces of $x$, where traces are generated by independently deleting each bit of $x$ with some deletion probability $q$. In this paper, we initiate the study of Circular trace reconstruction, where the unknown string $x$ is circular and traces are now rotated by a random cyclic shift. Trace reconstruction is related to many computational biology problems studying DNA, which is a primary motivation for this problem as well, as many types of DNA are known to be circular. Our main results are as follows. First, we prove that we can reconstruct arbitrary circular strings of length $n$ using $\exp\big(\tilde{O}(n^{1/3})\big)$ traces for any constant deletion probability $q$, as long as $n$ is prime or the product of two primes. For $n$ of this form, this nearly matches what was the best known bound of $\exp\big(O(n^{1/3})\big)$ for standard trace reconstruction when this paper was initially released. We note, however, that Chase very recently improved the standard trace reconstruction bound to $\exp\big(\tilde{O}(n^{1/5})\big)$. Next, we prove that we can reconstruct random circular strings with high probability using $n^{O(1)}$ traces for any constant deletion probability $q$. Finally, we prove a lower bound of $\tildeΩ(n^3)$ traces for arbitrary circular strings, which is greater than the best known lower bound of $\tildeΩ(n^{3/2})$ in standard trace reconstruction.

cs.DS↗

arXiv2020

Improved Algorithms for Population Recovery from the Deletion Channel

The population recovery problem asks one to recover an unknown distribution over $n$-bit strings given access to independent noisy samples of strings drawn from the distribution. Recently, Ban et al. [BCF+19] studied the problem where the noise is induced through the deletion channel. This problem generalizes the famous trace reconstruction problem, where one wishes to learn a single string under the deletion channel. Ban et al. showed how to learn $\ell$-sparse distributions over strings using $\exp\big(n^{1/2} \cdot (\log n)^{O(\ell)}\big)$ samples. In this work, we learn the distribution using only $\exp\big(\tilde{O}(n^{1/3}) \cdot \ell^2\big)$ samples, by developing a higher-moment analog of the algorithms of [DOS17, NP17], which solve trace reconstruction in $\exp\big(\tilde{O}(n^{1/3})\big)$ samples. We also give the first algorithm with a runtime subexponential in $n$, solving population recovery in $\exp\big(\tilde{O}(n^{1/3}) \cdot \ell^3\big)$ samples and time. Notably, our dependence on $n$ nearly matches the upper bound of [DOS17, NP17] when $\ell = O(1)$, and we reduce the dependence on $\ell$ from doubly to singly exponential. Therefore, we are able to learn large mixtures of strings: while Ban et al.'s algorithm can only learn a mixture of $O(\log n/\log \log n)$ strings with a subexponential number of samples, we are able to learn a mixture of $n^{o(1)}$ strings in $\exp\big(n^{1/3 + o(1)}\big)$ samples and time.

cs.DS↗

arXiv2020

On Distribution Testing in the Conditional Sampling Model

Recently, there has been significant work studying distribution testing under the Conditional Sampling model. In this model, a query specifies a subset $S$ of the domain, and the output received is a sample drawn from the distribution conditioned on being in $S$. In this paper, we improve query complexity bounds for several classic distribution testing problems in this model. First, we prove that tolerant uniformity testing in the conditional sampling model can be solved using $\tilde{O}(\varepsilon^{-2})$ queries, which is optimal and improves upon the $\tilde{O}(\varepsilon^{-20})$-query algorithm of Canonne et al. [CRS15]. This bound even holds under a restricted version of the conditional sampling model called the Pair Conditional Sampling model. Next, we prove that tolerant identity testing in the conditional sampling model can be solved in $\tilde{O}(\varepsilon^{-4})$ queries, which is the first known bound independent of the support size of the distribution for this problem. Next, we use our algorithm for tolerant uniformity testing to get an $\tilde{O}(\varepsilon^{-4})$-query algorithm for monotonicity testing in the conditional sampling model, improving on the $\tilde{O}(\varepsilon^{-22})$-query algorithm of Canonne [Can15]. Finally, we study (non-tolerant) identity testing under the pair conditional sampling model, and provide a tight bound of $\tildeΘ(\sqrt{\log N} \cdot \varepsilon^{-2})$ for the query complexity, where the domain of the distribution has size $N$. This improves upon both the known upper and lower bounds in [CRS15].

cs.DS↗

arXiv2020

3-wise Independent Random Walks can be Slightly Unbounded

Recently, many streaming algorithms have utilized generalizations of the fact that the expected maximum distance of any $4$-wise independent random walk on a line over $n$ steps is $O(\sqrt{n})$. In this paper, we show that $4$-wise independence is required for all of these algorithms, by constructing a $3$-wise independent random walk with expected maximum distance $Ω(\sqrt{n} \lg n)$ from the origin. We prove that this bound is tight for the first and second moment, and also extract a surprising matrix inequality from these results. Next, we consider a generalization where the steps $X_i$ are $k$-wise independent random variables with bounded $p$th moments. For general $k, p$, we determine the (asymptotically) maximum possible $p$th moment of the supremum of $X_1 + \dots + X_i$ over $1 \le i \le n$. We highlight the case $k = 4, p = 2$: here, we prove that the second moment of the furthest distance traveled is $O(\sum X_i^2)$. For this case, we only need the $X_i$'s to have bounded second moments and do not even need the $X_i$'s to be identically distributed. This implies an asymptotically stronger statement than Kolmogorov's maximal inequality that requires only $4$-wise independent random variables, and generalizes a recent result of Błasiok.

math.PR↗

arXiv2019

Functions on Antipower Prefix Lengths of the Thue-Morse Word

We say that a word $w$ of length $kn$ is a $k$-\textit{antipower} if it can be written in the form $w_1 \cdots w_k$, where each $w_i$ is a distinct word of length $n$. We analyze prefixes of the Thue-Morse word $\textbf{t}$ and lengths of antipowers occurring in them. Define $Γ(k)$ to be the largest odd $n$ such that the prefix of $\textbf{t}$ of length $kn$ is not a $k$-antipower, and $γ(k)$ to be the smallest odd $n$ such that the corresponding prefix is a $k$-antipower. We provide strong bounds on the asymptotic values of $γ(k)$ and $Γ(k)-γ(k)$. Our bounds on $γ(k)$ affirmatively answer one conjecture of Defant and make substantial progress towards answering a second conjecture of Defant. It was previously known that $Γ(k)$ and $γ(k)$ grow linearly in $k$, but our bounds on $Γ(k)-γ(k)$ prove that $Γ(k)-γ(k)$ also grows linearly in $k$.

math.CO↗

arXiv2019

Resolving Two Conjectures on Staircase Encodings and Boundary Grids of $132$ and $123$-avoiding permutations

This paper analyzes relations between pattern avoidance of certain permutations and graphs on staircase grids and boundary grids, and proves two conjectures posed by Bean, Tannock, and Ulfarsson (2015). More specifically, this paper enumerates a certain family of staircase encodings and proves that the downcore graph, a certain graph established on the boundary grid, is pure if and only if the permutation corresponding to the boundary grid avoids the classical patterns 123 and 2143.

math.CO↗

arXiv2018

Optimal terminal dimensionality reduction in Euclidean space

Let $\varepsilon\in(0,1)$ and $X\subset\mathbb R^d$ be arbitrary with $|X|$ having size $n>1$. The Johnson-Lindenstrauss lemma states there exists $f:X\rightarrow\mathbb R^m$ with $m = O(\varepsilon^{-2}\log n)$ such that $$ \forall x\in X\ \forall y\in X, \|x-y\|_2 \le \|f(x)-f(y)\|_2 \le (1+\varepsilon)\|x-y\|_2 . $$ We show that a strictly stronger version of this statement holds, answering one of the main open questions of [MMMR18]: "$\forall y\in X$" in the above statement may be replaced with "$\forall y\in\mathbb R^d$", so that $f$ not only preserves distances within $X$, but also distances to $X$ from the rest of space. Previously this stronger version was only known with the worse bound $m = O(\varepsilon^{-4}\log n)$. Our proof is via a tighter analysis of (a specific instantiation of) the embedding recipe of [MMMR18].

cs.DS↗

arXiv2018

The 26 Wilf-equivalence classes of length five quasi-consecutive patterns

We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$. We then prove additional equivalences to explicitly classify all quasi-consecutive patterns of length $5$ into 26 Wilf-equivalence classes.

math.CO↗

arXiv2018

Deterministic $O(1)$-Approximation Algorithms to 1-Center Clustering with Outliers

The 1-center clustering with outliers problem asks about identifying a prototypical robust statistic that approximates the location of a cluster of points. Given some constant $0 < α< 1$ and $n$ points such that $αn$ of them are in some (unknown) ball of radius $r,$ the goal is to compute a ball of radius $O(r)$ that also contains $αn$ points. This problem can be formulated with the points in a normed vector space such as $\mathbb{R}^d$ or in a general metric space. The problem has a simple randomized solution: a randomly selected point is a correct solution with constant probability, and its correctness can be verified in linear time. However, the deterministic complexity of this problem was not known. In this paper, for any $\ell_p$ vector space, we show an $O(nd)$-time solution with a ball of radius $O(r)$ for a fixed $α> \frac{1}{2},$ and for any normed vector space, we show an $O(nd)$-time solution with a ball of radius $O(r)$ when $α> \frac{1}{2}$ as well as an $O (nd \log^{(k)}(n))$-time solution with a ball of radius $O(r)$ for all $α> 0, k \in \mathbb{N},$ where $\log^{(k)}(n)$ represents the $k$th iterated logarithm, assuming distance computation and vector space operations take $O(d)$ time. For an arbitrary metric space, we show for any $C \in \mathbb{N}$ an $O(n^{1+1/C})$-time solution that finds a ball of radius $2Cr,$ assuming distance computation between any pair of points takes $O(1)$-time. Moreover, this algorithm is optimal for general metric spaces, as we show that for any fixed $α, C,$ there is no $o(n^{1+1/C})$-query and thus no $o(n^{1+1/C})$-time solution that deterministically finds a ball of radius $2Cr$.

cs.DS↗