26,896 research outputs found
Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Let C be a depth-3 circuit with n variables, degree d and top fanin k (called
sps(k,d,n) circuits) over base field F. It is a major open problem to design a
deterministic polynomial time blackbox algorithm that tests if C is identically
zero. Klivans & Spielman (STOC 2001) observed that the problem is open even
when k is a constant. This case has been subjected to a serious study over the
past few years, starting from the work of Dvir & Shpilka (STOC 2005).
We give the first polynomial time blackbox algorithm for this problem. Our
algorithm runs in time poly(nd^k), regardless of the base field. The only field
for which polynomial time algorithms were previously known is F=Q (Kayal &
Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first
blackbox algorithm for depth-3 circuits that does not use the rank based
approaches of Karnin & Shpilka (CCC 2008).
We prove an important tool for the study of depth-3 identities. We design a
blackbox polynomial time transformation that reduces the number of variables in
a sps(k,d,n) circuit to k variables, but preserves the identity structure.Comment: 14 pages, 1 figure, preliminary versio
Progress on Polynomial Identity Testing - II
We survey the area of algebraic complexity theory; with the focus being on
the problem of polynomial identity testing (PIT). We discuss the key ideas that
have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve
Algebraic Independence and Blackbox Identity Testing
Algebraic independence is an advanced notion in commutative algebra that
generalizes independence of linear polynomials to higher degree. Polynomials
{f_1, ..., f_m} \subset \F[x_1, ..., x_n] are called algebraically independent
if there is no non-zero polynomial F such that F(f_1, ..., f_m) = 0. The
transcendence degree, trdeg{f_1, ..., f_m}, is the maximal number r of
algebraically independent polynomials in the set. In this paper we design
blackbox and efficient linear maps \phi that reduce the number of variables
from n to r but maintain trdeg{\phi(f_i)}_i = r, assuming f_i's sparse and
small r. We apply these fundamental maps to solve several cases of blackbox
identity testing:
(1) Given a polynomial-degree circuit C and sparse polynomials f_1, ..., f_m
with trdeg r, we can test blackbox D := C(f_1, ..., f_m) for zeroness in
poly(size(D))^r time.
(2) Define a spsp_\delta(k,s,n) circuit C to be of the form \sum_{i=1}^k
\prod_{j=1}^s f_{i,j}, where f_{i,j} are sparse n-variate polynomials of degree
at most \delta. For k = 2 we give a poly(sn\delta)^{\delta^2} time blackbox
identity test.
(3) For a general depth-4 circuit we define a notion of rank. Assuming there
is a rank bound R for minimal simple spsp_\delta(k,s,n) identities, we give a
poly(snR\delta)^{Rk\delta^2} time blackbox identity test for spsp_\delta(k,s,n)
circuits. This partially generalizes the state of the art of depth-3 to depth-4
circuits.
The notion of trdeg works best with large or zero characteristic, but we also
give versions of our results for arbitrary fields.Comment: 32 pages, preliminary versio
An optimized energy potential can predict SH2 domain-peptide interactions
Peptide recognition modules (PRMs) are used throughout biology to mediate protein-protein interactions, and many PRMs are members of large protein domain families. Members of these families are often quite similar to each other, but each domain recognizes a distinct set of peptides, raising the question of how peptide recognition specificity is achieved using similar protein domains. The analysis of individual protein complex structures often gives answers that are not easily applicable to other members of the same PRM family. Bioinformatics-based approaches, one the other hand, may be difficult to interpret physically. Here we integrate structural information with a large, quantitative data set of SH2-peptide interactions to study the physical origin of domain-peptide specificity. We develop an energy model, inspired by protein folding, based on interactions between the amino acid positions in the domain and peptide. We use this model to successfully predict which SH2 domains and peptides interact and uncover the positions in each that are important for specificity. The energy model is general enough that it can be applied to other members of the SH2 family or to new peptides, and the cross-validation results suggest that these energy calculations will be useful for predicting binding interactions. It can also be adapted to study other PRM families, predict optimal peptides for a given SH2 domain, or study other biological interactions, e.g. protein-DNA interactions
Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains
We consider finite-state Markov chains that can be naturally decomposed into
smaller ``projection'' and ``restriction'' chains. Possibly this decomposition
will be inductive, in that the restriction chains will be smaller copies of the
initial chain. We provide expressions for Poincare (resp. log-Sobolev)
constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev)
constants of the projection and restriction chains, together with further a
parameter. In the case of the Poincare constant, our bound is always at least
as good as existing ones and, depending on the value of the extra parameter,
may be much better. There appears to be no previously published decomposition
result for the log-Sobolev constant. Our proofs are elementary and
self-contained.Comment: Published at http://dx.doi.org/10.1214/105051604000000639 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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