arXiv subjects

Igor Shinkar

Publications and source records attributed to Igor Shinkar.

At least 19 recordsLinked to original sources

arXiv2026

On Sampling Lower Bounds for Polynomials

In this work, we continue the line of research on the complexity of distributions (Viola, Journal of Computing 2012), and study samplers defined by low degree polynomials. An $n$-tuple $P = (P_1,\dots, P_n)$ of functions $P_i \colon \mathbb{F}_2^m \to \mathbb{F}_2$ defines a distribution over $\{0,1\}^n$ in the natural way: draw $X$ uniformly at random from $\mathbb{F}_2^m$ and output $(P_1(X),\dots, P_n(X)) \in \{0,1\}^n$. We show that when $P$ is defined by polynomials of degree $d$, the total variation distance of $P$ from the product distribution $\mathrm{Ber}(1/3)^{\otimes n}$ is $1-o_n(1)$, where $o_n(1)$ is a vanishing function of $n$ for any constant degree $d$. For small values of $d$, we show the following concrete bounds. (i) For $d=1$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(n))$. (ii) For $d=2$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(\log(n)/\log\log(n)))$. (iii) For $d=3$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(\sqrt{\log\log(n)}))$. Our results extend the recent lower bound results for sampling distributions, which have mostly focused on local samplers, small depth decision trees, and small depth circuits. As part of our proof, we establish the following result, that may be of independent interest: for any degree-$d$ polynomial $P\colon\mathbb{F}_2^m \to \mathbb{F}_2$ it holds that $\Pr_X[P(X) = 1]$ is bounded away from $1/3$ by some absolute constant $\delta = \delta_d>0$. Although the statement may seem obvious, we are not aware of an elementary proof of this. The proof techniques rely on the structural results for low degree polynomials, saying that any biased polynomial of degree $d$ can be written as a function of a small number of polynomials of degree $d-1$.

cs.CC↗

arXiv2024

Matrix Multiplication Reductions

In this paper we study a worst case to average case reduction for the problem of matrix multiplication over finite fields. Suppose we have an efficient average case algorithm, that given two random matrices $A,B$ outputs a matrix that has a non-trivial correlation with their product $A \cdot B$. Can we transform it into a worst case algorithm, that outputs the correct answer for all inputs without incurring a significant overhead in the running time? We present two results in this direction. (1) Two-sided error in the high agreement regime: We begin with a brief remark about a reduction for high agreement algorithms, i.e., an algorithm which agrees with the correct output on a large (say $>0.9$) fraction of entries, and show that the standard self-correction of linearity allows us to transform such algorithms into algorithms that work in worst case. (2) One-sided error in the low agreement regime: Focusing on average case algorithms with one-sided error, we show that over $\mathbb{F}_2$ there is a reduction that gets an $O(T)$ time average case algorithm that given a random input $A,B$ outputs a matrix that agrees with $A \cdot B$ on at least $51\%$ of the entries (i.e., has only a slight advantage over the trivial algorithm), and transforms it into an $\widetilde{O}(T)$ time worst case algorithm, that outputs the correct answer for all inputs with high probability.

cs.DS↗

arXiv2024

On the Power of Interactive Proofs for Learning

We continue the study of doubly-efficient proof systems for verifying agnostic PAC learning, for which we obtain the following results. - We construct an interactive protocol for learning the $t$ largest Fourier characters of a given function $f \colon \{0,1\}^n \to \{0,1\}$ up to an arbitrarily small error, wherein the verifier uses $\mathsf{poly}(t)$ random examples. This improves upon the Interactive Goldreich-Levin protocol of Goldwasser, Rothblum, Shafer, and Yehudayoff (ITCS 2021) whose sample complexity is $\mathsf{poly}(t,n)$. - For agnostically learning the class $\mathsf{AC}^0[2]$ under the uniform distribution, we build on the work of Carmosino, Impagliazzo, Kabanets, and Kolokolova (APPROX/RANDOM 2017) and design an interactive protocol, where given a function $f \colon \{0,1\}^n \to \{0,1\}$, the verifier learns the closest hypothesis up to $\mathsf{polylog}(n)$ multiplicative factor, using quasi-polynomially many random examples. In contrast, this class has been notoriously resistant even for constructing realisable learners (without a prover) using random examples. - For agnostically learning $k$-juntas under the uniform distribution, we obtain an interactive protocol, where the verifier uses $O(2^k)$ random examples to a given function $f \colon \{0,1\}^n \to \{0,1\}$. Crucially, the sample complexity of the verifier is independent of $n$. We also show that if we do not insist on doubly-efficient proof systems, then the model becomes trivial. Specifically, we show a protocol for an arbitrary class $\mathcal{C}$ of Boolean functions in the distribution-free setting, where the verifier uses $O(1)$ labeled examples to learn $f$.

cs.CC↗

arXiv2022

Quantum Worst-Case to Average-Case Reductions for All Linear Problems

We study the problem of designing worst-case to average-case reductions for quantum algorithms. For all linear problems, we provide an explicit and efficient transformation of quantum algorithms that are only correct on a small (even sub-constant) fraction of their inputs into ones that are correct on all inputs. This stands in contrast to the classical setting, where such results are only known for a small number of specific problems or restricted computational models. En route, we obtain a tight $\Omega(n^2)$ lower bound on the average-case quantum query complexity of the Matrix-Vector Multiplication problem. Our techniques strengthen and generalise the recently introduced additive combinatorics framework for classical worst-case to average-case reductions (STOC 2022) to the quantum setting. We rely on quantum singular value transformations to construct quantum algorithms for linear verification in superposition and learning Bogolyubov subspaces from noisy quantum oracles. We use these tools to prove a quantum local correction lemma, which lies at the heart of our reductions, based on a noise-robust probabilistic generalisation of Bogolyubov's lemma from additive combinatorics.

quant-ph↗

arXiv2022

Worst-Case to Average-Case Reductions via Additive Combinatorics

We present a new framework for designing worst-case to average-case reductions. For a large class of problems, it provides an explicit transformation of algorithms running in time $T$ that are only correct on a small (subconstant) fraction of their inputs into algorithms running in time $\widetilde{O}(T)$ that are correct on all inputs. Using our framework, we obtain such efficient worst-case to average-case reductions for fundamental problems in a variety of computational models; namely, algorithms for matrix multiplication, streaming algorithms for the online matrix-vector multiplication problem, and static data structures for all linear problems as well as for the multivariate polynomial evaluation problem. Our techniques crucially rely on additive combinatorics. In particular, we show a local correction lemma that relies on a new probabilistic version of the quasi-polynomial Bogolyubov-Ruzsa lemma.

cs.DS↗

arXiv2021

Derandomization of Cell Sampling

Since 1989, the best known lower bound on static data structures was Siegel's classical cell sampling lower bound. Siegel showed an explicit problem with $n$ inputs and $m$ possible queries such that every data structure that answers queries by probing $t$ memory cells requires space $s\geq\widetilde{\Omega}\left(n\cdot(\frac{m}{n})^{1/t}\right)$. In this work, we improve this bound for non-adaptive data structures to $s\geq\widetilde{\Omega}\left(n\cdot(\frac{m}{n})^{1/(t-1)}\right)$ for all $t \geq 2$. For $t=2$, we give a lower bound of $s>m-o(m)$, improving on the bound $s>m/2$ recently proved by Viola over $\mathbb{F}_2$ and Siegel's bound $s\geq\widetilde{\Omega}(\sqrt{mn})$ over other finite fields.

cs.DS↗

arXiv2020

Relaxed Locally Correctable Codes with Improved Parameters

Locally decodable codes (LDCs) are error-correcting codes $C : Σ^k \to Σ^n$ that admit a local decoding algorithm that recovers each individual bit of the message by querying only a few bits from a noisy codeword. An important question in this line of research is to understand the optimal trade-off between the query complexity of LDCs and their block length. Despite importance of these objects, the best known constructions of constant query LDCs have super-polynomial length, and there is a significant gap between the best constructions and the known lower bounds in terms of the block length. For many applications it suffices to consider the weaker notion of relaxed LDCs (RLDCs), which allows the local decoding algorithm to abort if by querying a few bits it detects that the input is not a codeword. This relaxation turned out to allow decoding algorithms with constant query complexity for codes with almost linear length. Specifically, [BGH+06] constructed an $O(q)$-query RLDC that encodes a message of length $k$ using a codeword of block length $n = O(k^{1+1/\sqrt{q}})$. In this work we improve the parameters of [BGH+06] by constructing an $O(q)$-query RLDC that encodes a message of length $k$ using a codeword of block length $O(k^{1+1/{q}})$. This construction matches (up to a multiplicative constant factor) the lower bounds of [KT00, Woo07] for constant query LDCs, thus making progress toward understanding the gap between LDCs and RLDCs in the constant query regime. In fact, our construction extends to the stronger notion of relaxed locally correctable codes (RLCCs), introduced in [GRR18], where given a noisy codeword the correcting algorithm either recovers each individual bit of the codeword by only reading a small part of the input, or aborts if the input is detected to be corrupt.

cs.CC↗

arXiv2020

Toward Probabilistic Checking against Non-Signaling Strategies with Constant Locality

Non-signaling strategies are a generalization of quantum strategies that have been studied in physics over the past three decades. Recently, they have found applications in theoretical computer science, including to proving inapproximability results for linear programming and to constructing protocols for delegating computation. A central tool for these applications is probabilistically checkable proof (PCPs) systems that are sound against non-signaling strategies. In this paper we show, assuming a certain geometrical hypothesis about noise robustness of non-signaling proofs (or, equivalently, about robustness to noise of solutions to the Sherali-Adams linear program), that a slight variant of the parallel repetition of the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound against non-signaling strategies with constant locality. Our proof relies on the analysis of the linearity test and agreement test (also known as the direct product test) in the non-signaling setting.

cs.CC↗

arXiv2019

Meyniel Extremal Families of Abelian Cayley Graphs

We study the game of Cops and Robbers, where cops try to capture a robber on the vertices of a graph. Meyniel's conjecture states that for every connected graph $G$ on $n$ vertices, the cop number of $G$ is upper bounded by $O(\sqrt{n})$, i.e., that $O(\sqrt{n})$ suffice to catch the robber. We present several families of abelian Cayley graphs that are Meyniel extremal, i.e., graphs whose cop number is $O(\sqrt{n})$. This proves that the $O(\sqrt{n})$ upper bound for Cayley graphs proved by Bradshaw is tight up to a multiplicative constant. In particular, this shows that Meyniel's conjecture, if true, is tight to a multiplicative constant even for abelian Cayley graphs. In order to prove the result, we construct Cayley graphs on $n$ vertices with $Ω(\sqrt{n})$ generators that are $K_{2,3}$-free. This shows that the Kövári, Sós, and Turán theorem, stating that any $K_{2,3}$-free graph of $n$ vertices has at most $O(n^{3/2})$ edges, is tight up to a multiplicative constant even for abelian Cayley graphs.

math.CO↗

arXiv2019

On Mappings on the Hypercube with Small Average Stretch

Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$, and let $\phi \colon \{0,1\}^{n-1} \to A$ be a bijection. We define the average stretch of $\phi$ as ${\sf avgStretch}(\phi) = {\mathbb E}[{\sf dist}(\phi(x),\phi(x'))]$, where the expectation is taken over uniformly random $x,x' \in \{0,1\}^{n-1}$ that differ in exactly one coordinate. In this paper we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results. (1) For any set $A \subseteq \{0,1\}^n$ of density $1/2$ there exists a bijection $\phi_A \colon \{0,1\}^{n-1} \to A$ such that ${\sf avgstretch}(\phi_A) = O(\sqrt{n})$. (2) For $n = 3^k$ let $A_{{\sf rec\text{-}maj}} = \{x \in \{0,1\}^n : {\sf rec\text{-}maj}(x) = 1\}$, where ${\sf rec\text{-}maj} : \{0,1\}^n \to \{0,1\}$ is the function recursive majority of 3's. There exists a bijection $\phi_{{\sf rec\text{-}maj}} \colon \{0,1\}^{n-1} \to A_{\sf rec\text{-}maj}$ such that ${\sf avgstretch}(\phi_{\sf rec\text{-}maj}) = O(1)$. (3) Let $A_{\sf tribes} = \{x \in \{0,1\}^n : {\sf tribes}(x) = 1\}$. There exists a bijection $\phi_{{\sf tribes}} \colon \{0,1\}^{n-1} \to A_{\sf tribes}$ such that ${\sf avgstretch}(\phi_{{\sf tribes}}) = O(\log(n))$. These results answer the questions raised by Benjamini et al.\ (FOCS 2014).

math.CO↗

arXiv2019

String Matching: Communication, Circuits, and Learning

String matching is the problem of deciding whether a given $n$-bit string contains a given $k$-bit pattern. We study the complexity of this problem in three settings. Communication complexity. For small $k$, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large $k$, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol. Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small $k$. Learning. We consider the problem of learning a hidden pattern of length at most $k$ relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem.

cs.CC↗

arXiv2018

Multitasking Capacity: Hardness Results and Improved Constructions

We consider the problem of determining the maximal $α\in (0,1]$ such that every matching $M$ of size $k$ (or at most $k$) in a bipartite graph $G$ contains an induced matching of size at least $α|M|$. This measure was recently introduced in Alon et al. (NIPS 2018) and is motivated by connectionist models of cognition as well as modeling interference in wireless and communication networks. We prove various hardness results for computing $α$ either exactly or approximately. En route to our results, we also consider the maximum connected matching problem: determining the largest matching $N$ in a graph $G$ such that every two edges in $N$ are connected by an edge. We prove a nearly optimal $n^{1-ε}$ hardness of approximation result (under randomized reductions) for connected matching in bipartite graphs (with both sides of cardinality $n$). Towards this end we define bipartite half-covers: A new combinatorial object that may be of independent interest. To the best of our knowledge, the best previous hardness result for the connected matching problem was some constant $β>1$. Finally, we demonstrate the existence of bipartite graphs with $n$ vertices on each side of average degree $d$, that achieve $α=1/2-ε$ for matchings of size sufficiently smaller than $n/poly(d)$. This nearly matches the trivial upper bound of $1/2$ on $α$ which holds for any graph containing a path of length 3.

cs.DS↗

arXiv2018

On Coloring Random Subgraphs of a Fixed Graph

Given an arbitrary graph $G$ we study the chromatic number of a random subgraph $G_{1/2}$ obtained from $G$ by removing each edge independently with probability $1/2$. Studying $χ(G_{1/2})$ has been suggested by Bukh~\cite{Bukh}, who asked whether $\mathbb{E}[χ(G_{1/2})] \geq Ω( χ(G)/\log(χ(G)))$ holds for all graphs $G$. In this paper we show that for any graph $G$ with chromatic number $k = χ(G)$ and for all $d \leq k^{1/3}$ it holds that $\Pr[χ(G_{1/2}) \leq d] < \exp \left(- Ω\left(\frac{k(k-d^3)}{d^3}\right)\right)$. In particular, $\Pr[G_{1/2} \text{ is bipartite}] < \exp \left(- Ω\left(k^2 \right)\right)$. The later bound is tight up to a constant in $Ω(\cdot)$, and is attained when $G$ is the complete graph on $k$ vertices. As a technical lemma, that may be of independent interest, we prove that if in \emph{any} $d^3$ coloring of the vertices of $G$ there are at least $t$ monochromatic edges, then $\Pr[χ(G_{1/2}) \leq d] < e^{- Ω\left(t\right)}$. We also prove that for any graph $G$ with chromatic number $k = χ(G)$ and independence number $α(G) \leq O(n/k)$ it holds that $\mathbb{E}[χ(G_{1/2})] \geq Ω\left( k/\log(k) \right)$. This gives a positive answer to the question of Bukh for a large family of graphs.

math.CO↗

arXiv2018

An Entropy Lower Bound for Non-Malleable Extractors

A $(k,\varepsilon)$-non-malleable extractor is a function ${\sf nmExt} : \{0,1\}^n \times \{0,1\}^d \to \{0,1\}$ that takes two inputs, a weak source $X \sim \{0,1\}^n$ of min-entropy $k$ and an independent uniform seed $s \in \{0,1\}^d$, and outputs a bit ${\sf nmExt}(X, s)$ that is $\varepsilon$-close to uniform, even given the seed $s$ and the value ${\sf nmExt}(X, s')$ for an adversarially chosen seed $s' \neq s$. Dodis and Wichs~(STOC 2009) showed the existence of $(k, \varepsilon)$-non-malleable extractors with seed length $d = \log(n-k-1) + 2\log(1/\varepsilon) + 6$ that support sources of entropy $k > \log(d) + 2 \log(1/\varepsilon) + 8$. We show that the foregoing bound is essentially tight, by proving that any $(k,\varepsilon)$-non-malleable extractor must satisfy the entropy bound $k > \log(d) + 2 \log(1/\varepsilon) - \log\log(1/\varepsilon) - C$ for an absolute constant $C$. In particular, this implies that non-malleable extractors require min-entropy at least $Ω(\log\log(n))$. This is in stark contrast to the existence of strong seeded extractors that support sources of entropy $k = O(\log(1/\varepsilon))$. Our techniques strongly rely on coding theory. In particular, we reveal an inherent connection between non-malleable extractors and error correcting codes, by proving a new lemma which shows that any $(k,\varepsilon)$-non-malleable extractor with seed length $d$ induces a code $C \subseteq \{0,1\}^{2^k}$ with relative distance $0.5 - 2\varepsilon$ and rate $\frac{d-1}{2^k}$.

cs.CC↗

arXiv2017

A Graph-Theoretic Approach to Multitasking

A key feature of neural network architectures is their ability to support the simultaneous interaction among large numbers of units in the learning and processing of representations. However, how the richness of such interactions trades off against the ability of a network to simultaneously carry out multiple independent processes -- a salient limitation in many domains of human cognition -- remains largely unexplored. In this paper we use a graph-theoretic analysis of network architecture to address this question, where tasks are represented as edges in a bipartite graph $G=(A \cup B, E)$. We define a new measure of multitasking capacity of such networks, based on the assumptions that tasks that \emph{need} to be multitasked rely on independent resources, i.e., form a matching, and that tasks \emph{can} be multitasked without interference if they form an induced matching. Our main result is an inherent tradeoff between the multitasking capacity and the average degree of the network that holds \emph{regardless of the network architecture}. These results are also extended to networks of depth greater than $2$. On the positive side, we demonstrate that networks that are random-like (e.g., locally sparse) can have desirable multitasking properties. Our results shed light into the parallel-processing limitations of neural systems and provide insights that may be useful for the analysis and design of parallel architectures.

cs.DM↗

arXiv2016

Sorting Networks On Restricted Topologies

The sorting number of a graph with $n$ vertices is the minimum depth of a sorting network with $n$ inputs and outputs that uses only the edges of the graph to perform comparisons. Many known results on sorting networks can be stated in terms of sorting numbers of different classes of graphs. In this paper we show the following general results about the sorting number of graphs. Any $n$-vertex graph that contains a simple path of length $d$ has a sorting network of depth $O(n \log(n/d))$. Any $n$-vertex graph with maximal degree $\Delta$ has a sorting network of depth $O(\Delta n)$. We also provide several results that relate the sorting number of a graph with its routing number, size of its maximal matching, and other well known graph properties. Additionally, we give some new bounds on the sorting number for some typical graphs.

cs.DS↗

arXiv2016

An $\widetilde{O}(n)$ Queries Adaptive Tester for Unateness

We present an adaptive tester for the unateness property of Boolean functions. Given a function $f:\{0,1\}^n \to \{0,1\}$ the tester makes $O(n \log(n)/ε)$ adaptive queries to the function. The tester always accepts a unate function, and rejects with probability at least 0.9 if a function is $ε$-far from being unate.

cs.DS↗

arXiv2015

A Counterexample to Monotonicity of Relative Mass in Random Walks

For a finite undirected graph $G = (V,E)$, let $p_{u,v}(t)$ denote the probability that a continuous-time random walk starting at vertex $u$ is in $v$ at time $t$. In this note we give an example of a Cayley graph $G$ and two vertices $u,v \in G$ for which the function \[ r_{u,v}(t) = \frac{p_{u,v}(t)}{p_{u,u}(t)} \qquad t \geq 0 \] is not monotonically non-decreasing. This answers a question asked by Peres in 2013.

math.PR↗