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Randomness Conductors and Constant-Degree Lossless Expanders [Extended Abstract]
The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible: (1-[epsilon])D, where D is the degree and [epsilon] is an arbitrarily small constant. The best previous explicit constructions gave expansion factor D/2, which is too weak for many applications. The D/2 bound was obtained via the eigenvalue method, and is known that that method cannot give better bounds.
The main abstract contribution of this paper is the introduction and initial study of randomness conductors, a notion which generalizes extractors, expanders, condensers and other similar objects. In all these functions, certain guarantee on the input "entropy" is converted to a guarantee on the output "entropy". For historical reasons, specific objects used specific guarantees of different flavors. We show that the flexibility afforded by the conductor definition leads to interesting combinations of these objects, and to better constructions such as those above. The main technical tool in these constructions is a natural generalization to conductors of the zig-zag graph product, previously defined for expanders and extractors.Engineering and Applied Science
Derandomization and Group Testing
The rapid development of derandomization theory, which is a fundamental area
in theoretical computer science, has recently led to many surprising
applications outside its initial intention. We will review some recent such
developments related to combinatorial group testing. In its most basic setting,
the aim of group testing is to identify a set of "positive" individuals in a
population of items by taking groups of items and asking whether there is a
positive in each group.
In particular, we will discuss explicit constructions of optimal or
nearly-optimal group testing schemes using "randomness-conducting" functions.
Among such developments are constructions of error-correcting group testing
schemes using randomness extractors and condensers, as well as threshold group
testing schemes from lossless condensers.Comment: Invited Paper in Proceedings of 48th Annual Allerton Conference on
Communication, Control, and Computing, 201
Better short-seed quantum-proof extractors
We construct a strong extractor against quantum storage that works for every
min-entropy , has logarithmic seed length, and outputs bits,
provided that the quantum adversary has at most qubits of memory, for
any \beta < \half. The construction works by first condensing the source
(with minimal entropy-loss) and then applying an extractor that works well
against quantum adversaries when the source is close to uniform.
We also obtain an improved construction of a strong quantum-proof extractor
in the high min-entropy regime. Specifically, we construct an extractor that
uses a logarithmic seed length and extracts bits from any source
over \B^n, provided that the min-entropy of the source conditioned on the
quantum adversary's state is at least , for any \beta < \half.Comment: 14 page
Better lossless condensers through derandomized curve samplers
Lossless condensers are unbalanced expander graphs, with expansion close to optimal. Equivalently, they may be viewed as functions that use a short random seed to map a source on n bits to a source on many fewer bits while preserving all of the min-entropy. It is known how to build lossless condensers when the graphs are slightly unbalanced in the work of M. Capalbo et al. (2002). The highly unbalanced case is also important but the only known construction does not condense the source well. We give explicit constructions of lossless condensers with condensing close to optimal, and using near-optimal seed length. Our main technical contribution is a randomness-efficient method for sampling FD (where F is a field) with low-degree curves. This problem was addressed before in the works of E. Ben-Sasson et al. (2003) and D. Moshkovitz and R. Raz (2006) but the solutions apply only to degree one curves, i.e., lines. Our technique is new and elegant. We use sub-sampling and obtain our curve samplers by composing a sequence of low-degree manifolds, starting with high-dimension, low-degree manifolds and proceeding through lower and lower dimension manifolds with (moderately) growing degrees, until we finish with dimension-one, low-degree manifolds, i.e., curves. The technique may be of independent interest
Vanishingly Sparse Matrices and Expander Graphs, With Application to Compressed Sensing
We revisit the probabilistic construction of sparse random matrices where
each column has a fixed number of nonzeros whose row indices are drawn
uniformly at random with replacement. These matrices have a one-to-one
correspondence with the adjacency matrices of fixed left degree expander
graphs. We present formulae for the expected cardinality of the set of
neighbors for these graphs, and present tail bounds on the probability that
this cardinality will be less than the expected value. Deducible from these
bounds are similar bounds for the expansion of the graph which is of interest
in many applications. These bounds are derived through a more detailed analysis
of collisions in unions of sets. Key to this analysis is a novel {\em dyadic
splitting} technique. The analysis led to the derivation of better order
constants that allow for quantitative theorems on existence of lossless
expander graphs and hence the sparse random matrices we consider and also
quantitative compressed sensing sampling theorems when using sparse non
mean-zero measurement matrices.Comment: 17 pages, 12 Postscript figure
Noise-Resilient Group Testing: Limitations and Constructions
We study combinatorial group testing schemes for learning -sparse Boolean
vectors using highly unreliable disjunctive measurements. We consider an
adversarial noise model that only limits the number of false observations, and
show that any noise-resilient scheme in this model can only approximately
reconstruct the sparse vector. On the positive side, we take this barrier to
our advantage and show that approximate reconstruction (within a satisfactory
degree of approximation) allows us to break the information theoretic lower
bound of that is known for exact reconstruction of
-sparse vectors of length via non-adaptive measurements, by a
multiplicative factor .
Specifically, we give simple randomized constructions of non-adaptive
measurement schemes, with measurements, that allow efficient
reconstruction of -sparse vectors up to false positives even in the
presence of false positives and false negatives within the
measurement outcomes, for any constant . We show that, information
theoretically, none of these parameters can be substantially improved without
dramatically affecting the others. Furthermore, we obtain several explicit
constructions, in particular one matching the randomized trade-off but using measurements. We also obtain explicit constructions
that allow fast reconstruction in time \poly(m), which would be sublinear in
for sufficiently sparse vectors. The main tool used in our construction is
the list-decoding view of randomness condensers and extractors.Comment: Full version. A preliminary summary of this work appears (under the
same title) in proceedings of the 17th International Symposium on
Fundamentals of Computation Theory (FCT 2009
Efficient and Robust Compressed Sensing Using Optimized Expander Graphs
Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any n-dimensional vector that is k-sparse can be fully recovered using O(klog n) measurements and only O(klog n) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(nlog(n/k))). We also show that by tolerating a small penal- ty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k-sparse signal and then, using very simple optimization techniques, finds a k-sparse signal which is close to the best k-term approximation of the original signal
Extracting Mergers and Projections of Partitions
We study the problem of extracting randomness from somewhere-random sources,
and related combinatorial phenomena: partition analogues of Shearer's lemma on
projections.
A somewhere-random source is a tuple of (possibly
correlated) -valued random variables where for some unknown , is guaranteed to be uniformly distributed. An
is a seeded device that takes a somewhere-random source as input and
outputs nearly uniform random bits. We study the seed-length needed for
extracting mergers with constant and constant error. We show:
Just like in the case of standard extractors, seedless extracting
mergers with even just one output bit do not exist.
Unlike the case of standard extractors, it possible to have
extracting mergers that output a constant number of bits using only constant
seed. Furthermore, a random choice of merger does not work for this purpose!
Nevertheless, just like in the case of standard extractors, an
extracting merger which gets most of the entropy out (namely, having
output bits) must have seed. This is the main
technical result of our work, and is proved by a second-moment strengthening of
the graph-theoretic approach of Radhakrishnan and Ta-Shma to extractors.
In contrast, seed-length/output-length tradeoffs for condensing mergers
(where the output is only required to have high min-entropy), can be fully
explained by using standard condensers.
Inspired by such considerations, we also formulate a new and basic class of
problems in combinatorics: partition analogues of Shearer's lemma. We show
basic results in this direction; in particular, we prove that in any partition
of the -dimensional cube into two parts, one of the parts has an
axis parallel -dimensional projection of area at least .Comment: Full version of the paper accepted to the International Conference on
Randomization and Computation (RANDOM) 2023. 28 pages, 2 figure
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