Dean Doron

On the Problem of Approximating the Eigenvalues of Undirected Graphs in Probabilistic Logspace

We introduce the problem of approximating the eigenvalues of a given stochastic/symmetric matrix in the context of classical space-bounded computation.

On Approximating the Eigenvalues of Stochastic Matrices in Probabilistic Logspace

We show that approximating the second eigenvalue of stochastic operators is BPL-complete, thus giving a natural problem complete for this class. We also show that approximating any eigenvalue of a stochastic and Hermitian operator with constant accuracy can be done in BPL. This work together with related work on the subject reveal a picture where the various space-bounded classes (e.g., probabilistic logspace, quantum logspace and the class DET) can be characterized by algebraic problems (such as approximating the spectral gap) where, roughly speaking, the difference between the classes lies in the kind of operators they can handle (e.g., stochastic, Hermitian or arbitrary).

An efficient reduction from two-source to non-malleable extractors: achieving near-logarithmic min-entropy

The breakthrough result of Chattopadhyay and Zuckerman (2016) gives a reduction from the construction of explicit two-source extractors to the construction of explicit non-malleable extractors. However, even assuming the existence of optimal explicit non-malleable extractors only gives a two-source extractor (or a Ramsey graph) for poly(logn) entropy, rather than the optimal O(logn). In this paper we modify the construction to solve the above barrier. Using the currently best explicit non-malleable extractors we get an explicit bipartite Ramsey graphs for sets of size 2k, for k=O(logn loglogn). Any further improvement in the construction of non-malleable extractors would immediately yield a corresponding two-source extractor. Intuitively, Chattopadhyay and Zuckerman use an extractor as a sampler, and we observe that one could use a weaker object - a somewhere-random condenser with a small entropy gap and a very short seed. We also show how to explicitly construct this weaker object using the error reduction technique of Raz, Reingold and Vadhan (1999), and the constant-degree dispersers of Zuckerman (2006) that also work against extremely small tests.

On the de-randomization of space-bounded approximate counting problems

Revisiting the problem of de-randomizing approximate counting.We study approximation problems in the space-bounded model motivated by quantum algorithms for linear algebraic problems.Deterministic logspace approximation schemes under de-randomization assumptions.Randomized logspace approximation schemes under de-quantumization assumptions. It was recently shown that SVD and matrix inversion can be approximated in quantum log-space 1] for well formed matrices. This can be interpreted as a fully logarithmic quantum approximation scheme for both problems. We show that if prBQL = prBPL then every fully logarithmic quantum approximation scheme can be replaced by a probabilistic one. Hence, if classical algorithms cannot approximate the above functions in logarithmic space, then there is a gap already for languages, namely, prBQL ? prBPL .On the way we simplify a proof of Goldreich for a similar statement for time bounded probabilistic algorithms. We show that our simplified algorithm works also in the space bounded setting (for a large set of functions) whereas Goldreichs approach does not seem to apply in the space bounded setting.

A new approach for constructing low-error, two-source extractors

Our main contribution in this paper is a new reduction from explicit two-source extractors for polynomially-small entropy rate and negligible error to explicit t-non-malleable extractors with seed-length that has a good dependence on t. Our reduction is based on the Chattopadhyay and Zuckerman framework (STOC 2016), and surprisingly we dispense with the use of resilient functions which appeared to be a major ingredient there and in follow-up works. The use of resilient functions posed a fundamental barrier towards achieving negligible error, and our new reduction circumvents this bottleneck. The parameters we require from t-non-malleable extractors for our reduction to work hold in a non-explicit construction, but currently it is not known how to explicitly construct such extractors. As a result we do not give an unconditional construction of an explicit low-error two-source extractor. Nonetheless, we believe our work gives a viable approach for solving the important problem of low-error two-source extractors. Furthermore, our work highlights an existing barrier in constructing low-error two-source extractors, and draws attention to the dependence of the parameter t in the seed-length of the non-malleable extractor. We hope this work would lead to further developments in explicit constructions of both non-malleable and two-source extractors.