arXiv subjects

Pritish Kamath

Publications and source records attributed to Pritish Kamath.

59 records · Page 4Linked to original sources

arXiv2017

Improved Bounds for Universal One-Bit Compressive Sensing

Unlike compressive sensing where the measurement outputs are assumed to be real-valued and have infinite precision, in "one-bit compressive sensing", measurements are quantized to one bit, their signs. In this work, we show how to recover the support of sparse high-dimensional vectors in the one-bit compressive sensing framework with an asymptotically near-optimal number of measurements. We also improve the bounds on the number of measurements for approximately recovering vectors from one-bit compressive sensing measurements. Our results are universal, namely the same measurement scheme works simultaneously for all sparse vectors. Our proof of optimality for support recovery is obtained by showing an equivalence between the task of support recovery using 1-bit compressive sensing and a well-studied combinatorial object known as Union Free Families.

cs.IT↗

arXiv2016

Decidability of Non-Interactive Simulation of Joint Distributions

We present decidability results for a sub-class of "non-interactive" simulation problems, a well-studied class of problems in information theory. A non-interactive simulation problem is specified by two distributions $P(x,y)$ and $Q(u,v)$: The goal is to determine if two players, Alice and Bob, that observe sequences $X^n$ and $Y^n$ respectively where $\{(X_i, Y_i)\}_{i=1}^n$ are drawn i.i.d. from $P(x,y)$ can generate pairs $U$ and $V$ respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to $Q(u,v)$. Even when $P$ and $Q$ are extremely simple: e.g., $P$ is uniform on the triples $\{(0,0), (0,1), (1,0)\}$ and $Q$ is a "doubly symmetric binary source", i.e., $U$ and $V$ are uniform $\pm 1$ variables with correlation say $0.49$, it is open if $P$ can simulate $Q$. In this work, we show that whenever $P$ is a distribution on a finite domain and $Q$ is a $2 \times 2$ distribution, then the non-interactive simulation problem is decidable: specifically, given $δ> 0$ the algorithm runs in time bounded by some function of $P$ and $δ$ and either gives a non-interactive simulation protocol that is $δ$-close to $Q$ or asserts that no protocol gets $O(δ)$-close to $Q$. The main challenge to such a result is determining explicit (computable) convergence bounds on the number $n$ of samples that need to be drawn from $P(x,y)$ to get $δ$-close to $Q$. We invoke contemporary results from the analysis of Boolean functions such as the invariance principle and a regularity lemma to obtain such explicit bounds.

cs.IT↗

arXiv2015

Communication Complexity of Permutation-Invariant Functions

Motivated by the quest for a broader understanding of communication complexity of simple functions, we introduce the class of "permutation-invariant" functions. A partial function $f:\{0,1\}^n \times \{0,1\}^n\to \{0,1,?\}$ is permutation-invariant if for every bijection $π:\{1,\ldots,n\} \to \{1,\ldots,n\}$ and every $\mathbf{x}, \mathbf{y} \in \{0,1\}^n$, it is the case that $f(\mathbf{x}, \mathbf{y}) = f(\mathbf{x}^π, \mathbf{y}^π)$. Most of the commonly studied functions in communication complexity are permutation-invariant. For such functions, we present a simple complexity measure (computable in time polynomial in $n$ given an implicit description of $f$) that describes their communication complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the communication complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as 'Set-Disjointness' and 'Indexing', while complementing them with the relatively lesser-known upper bounds for 'Gap-Inner-Product' (from the sketching literature) and 'Sparse-Gap-Inner-Product' (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of communication complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in communication complexity after an additive $O(\log \log n)$ overhead.

cs.CC↗

arXiv2012

Preservation under Substructures modulo Bounded Cores

We investigate a model-theoretic property that generalizes the classical notion of "preservation under substructures". We call this property \emph{preservation under substructures modulo bounded cores}, and present a syntactic characterization via $Σ_2^0$ sentences for properties of arbitrary structures definable by FO sentences. As a sharper characterization, we further show that the count of existential quantifiers in the $Σ_2^0$ sentence equals the size of the smallest bounded core. We also present our results on the sharper characterization for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the sharper characterization fails, but for which the classical Łoś-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the Łoś-Tarski theorem for some of the aforementioned cases.

cs.LO↗

arXiv2012

Faster Algorithms for Alternating Refinement Relations

One central issue in the formal design and analysis of reactive systems is the notion of refinement that asks whether all behaviors of the implementation is allowed by the specification. The local interpretation of behavior leads to the notion of simulation. Alternating transition systems (ATSs) provide a general model for composite reactive systems, and the simulation relation for ATSs is known as alternating simulation. The simulation relation for fair transition systems is called fair simulation. In this work our main contributions are as follows: (1) We present an improved algorithm for fair simulation with Büchi fairness constraints; our algorithm requires $O(n^3 \cdot m)$ time as compared to the previous known $O(n^6)$-time algorithm, where $n$ is the number of states and $m$ is the number of transitions. (2) We present a game based algorithm for alternating simulation that requires $O(m^2)$-time as compared to the previous known $O((n \cdot m)^2)$-time algorithm, where $n$ is the number of states and $m$ is the size of transition relation. (3) We present an iterative algorithm for alternating simulation that matches the time complexity of the game based algorithm, but is more space efficient than the game based algorithm.

cs.LO↗