502 research outputs found
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Adaptive Boolean Monotonicity Testing in Total Influence Time
Testing monotonicity of a Boolean function f:{0,1}^n -> {0,1} is an important problem in the field of property testing. It has led to connections with many interesting combinatorial questions on the directed hypercube: routing, random walks, and new isoperimetric theorems. Denoting the proximity parameter by epsilon, the best tester is the non-adaptive O~(epsilon^{-2}sqrt{n}) tester of Khot-Minzer-Safra (FOCS 2015). A series of recent results by Belovs-Blais (STOC 2016) and Chen-Waingarten-Xie (STOC 2017) have led to Omega~(n^{1/3}) lower bounds for adaptive testers. Reducing this gap is a significant question, that touches on the role of adaptivity in monotonicity testing of Boolean functions.
We approach this question from the perspective of parametrized property testing, a concept recently introduced by Pallavoor-Raskhodnikova-Varma (ACM TOCT 2017), where one seeks to understand performance of testers with respect to parameters other than just the size. Our result is an adaptive monotonicity tester with one-sided error whose query complexity is O(epsilon^{-2}I(f)log^5 n), where I(f) is the total influence of the function. Therefore, adaptivity provably helps monotonicity testing for low influence functions
Learning to Navigate the Energy Landscape
In this paper, we present a novel and efficient architecture for addressing
computer vision problems that use `Analysis by Synthesis'. Analysis by
synthesis involves the minimization of the reconstruction error which is
typically a non-convex function of the latent target variables.
State-of-the-art methods adopt a hybrid scheme where discriminatively trained
predictors like Random Forests or Convolutional Neural Networks are used to
initialize local search algorithms. While these methods have been shown to
produce promising results, they often get stuck in local optima. Our method
goes beyond the conventional hybrid architecture by not only proposing multiple
accurate initial solutions but by also defining a navigational structure over
the solution space that can be used for extremely efficient gradient-free local
search. We demonstrate the efficacy of our approach on the challenging problem
of RGB Camera Relocalization. To make the RGB camera relocalization problem
particularly challenging, we introduce a new dataset of 3D environments which
are significantly larger than those found in other publicly-available datasets.
Our experiments reveal that the proposed method is able to achieve
state-of-the-art camera relocalization results. We also demonstrate the
generalizability of our approach on Hand Pose Estimation and Image Retrieval
tasks
The Point-to-Set Principle, the Continuum Hypothesis, and the Dimensions of Hamel Bases
We prove that the Continuum Hypothesis implies that every real number in
(0,1] is the Hausdorff dimension of a Hamel basis of the vector space of reals
over the field of rationals.
The logic of our proof is of particular interest. The statement of our
theorem is classical; it does not involve the theory of computing. However, our
proof makes essential use of algorithmic fractal dimension--a
computability-theoretic construct--and the point-to-set principle of J. Lutz
and N. Lutz (2018)
When Hashes Met Wedges: A Distributed Algorithm for Finding High Similarity Vectors
Finding similar user pairs is a fundamental task in social networks, with
numerous applications in ranking and personalization tasks such as link
prediction and tie strength detection. A common manifestation of user
similarity is based upon network structure: each user is represented by a
vector that represents the user's network connections, where pairwise cosine
similarity among these vectors defines user similarity. The predominant task
for user similarity applications is to discover all similar pairs that have a
pairwise cosine similarity value larger than a given threshold . In
contrast to previous work where is assumed to be quite close to 1, we
focus on recommendation applications where is small, but still
meaningful. The all pairs cosine similarity problem is computationally
challenging on networks with billions of edges, and especially so for settings
with small . To the best of our knowledge, there is no practical solution
for computing all user pairs with, say on large social networks,
even using the power of distributed algorithms.
Our work directly addresses this challenge by introducing a new algorithm ---
WHIMP --- that solves this problem efficiently in the MapReduce model. The key
insight in WHIMP is to combine the "wedge-sampling" approach of Cohen-Lewis for
approximate matrix multiplication with the SimHash random projection techniques
of Charikar. We provide a theoretical analysis of WHIMP, proving that it has
near optimal communication costs while maintaining computation cost comparable
with the state of the art. We also empirically demonstrate WHIMP's scalability
by computing all highly similar pairs on four massive data sets, and show that
it accurately finds high similarity pairs. In particular, we note that WHIMP
successfully processes the entire Twitter network, which has tens of billions
of edges
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