arXiv subjects

Subhabrata Sen

Publications and source records attributed to Subhabrata Sen.

49 records · Page 3Linked to original sources

arXiv2018

LBP: Robust Rate Adaptation Algorithm for SVC Video Streaming

Video streaming today accounts for up to 55\% of mobile traffic. In this paper, we explore streaming videos encoded using Scalable Video Coding scheme (SVC) over highly variable bandwidth conditions such as cellular networks. SVC's unique encoding scheme allows the quality of a video chunk to change incrementally, making it more flexible and adaptive to challenging network conditions compared to other encoding schemes. Our contribution is threefold. First, we formulate the quality decisions of video chunks constrained by the available bandwidth, the playback buffer, and the chunk deadlines as an optimization problem. The objective is to optimize a novel QoE metric that models a combination of the three objectives of minimizing the stall/skip duration of the video, maximizing the playback quality of every chunk, and minimizing the number of quality switches. Second, we develop Layered Bin Packing (LBP) Adaptation Algorithm, a novel algorithm that solves the proposed optimization problem. Moreover, we show that LBP achieves the optimal solution of the proposed optimization problem with linear complexity in the number of video chunks. Third, we propose an online algorithm (online LBP) where several challenges are addressed including handling bandwidth prediction errors, and short prediction duration. Extensive simulations with real bandwidth traces of public datasets reveal the robustness of our scheme and demonstrate its significant performance improvement as compared to the state-of-the-art SVC streaming algorithms. The proposed algorithm is also implemented on a TCP/IP emulation test bed with real LTE bandwidth traces, and the emulation confirms the simulation results and validates that the algorithm can be implemented and deployed on today's mobile devices.

cs.NI↗

arXiv2018

Preferential Attachment When Stable

We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $α^{th}$ power $(α>1)$ of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of $2n-2$ new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, a fine control on this probability may be leveraged to derive a lower tail Large Deviation Principle (LDP) for $L = \sum_{i=1}^{n} \frac{S_i^2}{i^2}$, where $\{S_n : n \geq 0\}$ is a simple symmetric random walk started at zero. We provide an alternate proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous time analogue of $L$. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

math.PR↗

arXiv2018

The threshold for SDP-refutation of random regular NAE-3SAT

Unlike its cousin 3SAT, the NAE-3SAT (not-all-equal-3SAT) problem has the property that spectral/SDP algorithms can efficiently refute random instances when the constraint density is a large constant (with high probability). But do these methods work immediately above the "satisfiability threshold", or is there still a range of constraint densities for which random NAE-3SAT instances are unsatisfiable but hard to refute? We show that the latter situation prevails, at least in the context of random regular instances and SDP-based refutation. More precisely, whereas a random $d$-regular instance of NAE-3SAT is easily shown to be unsatisfiable (whp) once $d \geq 8$, we establish the following sharp threshold result regarding efficient refutation: If $d < 13.5$ then the basic SDP, even augmented with triangle inequalities, fails to refute satisfiability (whp), if $d > 13.5$ then even the most basic spectral algorithm refutes satisfiability~(whp).

cs.DS↗

arXiv2017

Phase transitions of extremal cuts for the configuration model

The $k$-section width and the Max-Cut for the configuration model are shown to exhibit phase transitions according to the values of certain parameters of the asymptotic degree distribution. These transitions mirror those observed on Erdős-Rényi random graphs, established by Luczak and McDiarmid (2001), and Coppersmith et al. (2004), respectively.

math.CO↗

arXiv2017

Optimization on Sparse Random Hypergraphs and Spin Glasses

We establish that in the large degree limit, the value of certain optimization problems on sparse random hypergraphs is determined by an appropriate Gaussian optimization problem. This approach was initiated in Dembo et. al.(2016) for extremal cuts of graphs. The usefulness of this technique is further illustrated by deriving the optimal value for Max $q$-cut on Erdős-Rényi and random regular graphs, Max XORSAT on Erdős-Rényi hypergraphs, and the min-bisection for the Stochastic Block Model.

math.PR↗

arXiv2017

Detection Thresholds for the $β$-Model on Sparse Graphs

In this paper we study sharp thresholds for detecting sparse signals in $β$-models for potentially sparse random graphs. The results demonstrate interesting interplay between graph sparsity, signal sparsity, and signal strength. In regimes of moderately dense signals, irrespective of graph sparsity, the detection thresholds mirror corresponding results in independent Gaussian sequence problems. For sparser signals, extreme graph sparsity implies that all tests are asymptotically powerless, irrespective of the signal strength. On the other hand, sharp detection thresholds are obtained, up to matching constants, on denser graphs. The phase transition mentioned above are sharp. As a crucial ingredient, we study a version of the Higher Criticism Test which is provably sharp up to optimal constants in the regime of sparse signals. The theoretical results are further verified by numerical simulations.

math.ST↗

arXiv2017

A connection between MAX $κ$-CUT and the inhomogeneous Potts spin glass in the large degree limit

We study the asymptotic behavior of the Max $κ$-cut on a family of sparse, inhomogeneous random graphs. In the large degree limit, the leading term is a variational problem, involving the ground state of a constrained inhomogeneous Potts spin glass. We derive a Parisi type formula for the free energy of this model, with possible constraints on the proportions, and derive the limiting ground state energy by a suitable zero temperature limit.

math.PR↗

arXiv2016

Near-Optimal Disjoint-Path Facility Location Through Set Cover by Pairs

In this paper we consider two special cases of the "cover-by-pairs" optimization problem that arise when we need to place facilities so that each customer is served by two facilities that reach it by disjoint shortest paths. These problems arise in a network traffic monitoring scheme proposed by Breslau et al. and have potential applications to content distribution. The "set-disjoint" variant applies to networks that use the OSPF routing protocol, and the "path-disjoint" variant applies when MPLS routing is enabled, making better solutions possible at the cost of greater operational expense. Although we can prove that no polynomial-time algorithm can guarantee good solutions for either version, we are able to provide heuristics that do very well in practice on instances with real-world network structure. Fast implementations of the heuristics, made possible by exploiting mathematical observations about the relationship between the network instances and the corresponding instances of the cover-by-pairs problem, allow us to perform an extensive experimental evaluation of the heuristics and what the solutions they produce tell us about the effectiveness of the proposed monitoring scheme. For the set-disjoint variant, we validate our claim of near-optimality via a new lower-bounding integer programming formulation. Although computing this lower bound requires solving the NP-hard Hitting Set problem and can underestimate the optimal value by a linear factor in the worst case, it can be computed quickly by CPLEX, and it equals the optimal solution value for all the instances in our extensive testbed.

cs.DS↗

arXiv2016

Optimal Adaptive Inference in Random Design Binary Regression

We construct confidence sets for the regression function in nonparametric binary regression with an unknown design density. These confidence sets are adaptive in $L^2$ loss over a continuous class of Sobolev type spaces. Adaptation holds in the smoothness of the regression function, over the maximal parameter spaces where adaptation is possible, provided the design density is smooth enough. We identify two key regimes --- one where adaptation is possible, and one where some critical regions must be removed. We address related questions about goodness of fit testing and adaptive estimation of relevant parameters.

math.ST↗

arXiv2015

Semidefinite Programs on Sparse Random Graphs and their Application to Community Detection

Denote by $A$ the adjacency matrix of an Erdos-Renyi graph with bounded average degree. We consider the problem of maximizing $\langle A-E\{A\},X\rangle$ over the set of positive semidefinite matrices $X$ with diagonal entries $X_{ii}=1$. We prove that for large (bounded) average degree $d$, the value of this semidefinite program (SDP) is --with high probability-- $2n\sqrt{d} + n\, o(\sqrt{d})+o(n)$. For a random regular graph of degree $d$, we prove that the SDP value is $2n\sqrt{d-1}+o(n)$, matching a spectral upper bound. Informally, Erdos-Renyi graphs appear to behave similarly to random regular graphs for semidefinite programming. We next consider the sparse, two-groups, symmetric community detection problem (also known as planted partition). We establish that SDP achieves the information-theoretically optimal detection threshold for large (bounded) degree. Namely, under this model, the vertex set is partitioned into subsets of size $n/2$, with edge probability $a/n$ (within group) and $b/n$ (across). We prove that SDP detects the partition with high probability provided $(a-b)^2/(4d)> 1+o_{d}(1)$, with $d= (a+b)/2$. By comparison, the information theoretic threshold for detecting the hidden partition is $(a-b)^2/(4d)> 1$: SDP is nearly optimal for large bounded average degree. Our proof is based on tools from different research areas: $(i)$ A new `higher-rank' Grothendieck inequality for symmetric matrices; $(ii)$ An interpolation method inspired from statistical physics; $(iii)$ An analysis of the eigenvectors of deformed Gaussian random matrices.

cs.DM↗

arXiv2015

High Temperature Asymptotics of Orthogonal Mean-Field Spin Glasses

We evaluate the high temperature limit of the free energy of spin glasses on the hypercube with Hamiltonian $H_N(σ) = σ^T J σ$, where the coupling matrix $J$ is drawn from certain symmetric orthogonally invariant ensembles. Our derivation relates the annealed free energy of these models to a spherical integral, and expresses the limit of the free energy in terms of the limiting spectral measure of the coupling matrix $J$. As an application, we derive the limiting free energy of the Random Orthogonal Model (ROM) at high temperatures, which confirms non-rigorous calculations of Marinari et al. (1994). Our methods also apply to other well-known models of disordered systems, including the SK and Gaussian Hopfield models.

math.PR↗

arXiv2015

Extremal Cuts of Sparse Random Graphs

For Erdős-Rényi random graphs with average degree $γ$, and uniformly random $γ$-regular graph on $n$ vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are $n\Big(\fracγ{4} + {\sf P}_* \sqrt{\fracγ{4}} + o(\sqrtγ)\Big) + o(n)$ while the size of the minimum bisection is $n\Big(\fracγ{4}-{\sf P}_*\sqrt{\fracγ{4}} + o(\sqrtγ)\Big) + o(n)$. Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington-Kirkpatrick model, with ${\sf P}_* \approx 0.7632$ standing for the ground state energy of the latter, expressed analytically via Parisi's formula.

math.PR↗

arXiv2010

Coordinated Weighted Sampling for Estimating Aggregates Over Multiple Weight Assignments

Many data sources are naturally modeled by multiple weight assignments over a set of keys: snapshots of an evolving database at multiple points in time, measurements collected over multiple time periods, requests for resources served at multiple locations, and records with multiple numeric attributes. Over such vector-weighted data we are interested in aggregates with respect to one set of weights, such as weighted sums, and aggregates over multiple sets of weights such as the $L_1$ difference. Sample-based summarization is highly effective for data sets that are too large to be stored or manipulated. The summary facilitates approximate processing queries that may be specified after the summary was generated. Current designs, however, are geared for data sets where a single {\em scalar} weight is associated with each key. We develop a sampling framework based on {\em coordinated weighted samples} that is suited for multiple weight assignments and obtain estimators that are {\em orders of magnitude tighter} than previously possible. We demonstrate the power of our methods through an extensive empirical evaluation on diverse data sets ranging from IP network to stock quotes data.

cs.DB↗