927 research outputs found
Algebraic and Combinatorial Methods in Computational Complexity
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings
Fourier Growth of Structured ??-Polynomials and Applications
We analyze the Fourier growth, i.e. the L? Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F?-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators.
Our main structural results on Fourier growth are as follows:
- We show that any symmetric degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013].
- We show that any read-? degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? (k ? d)^{O(k)}.
- We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds.
Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F?-polynomials
Directional States of Symmetric-Top Molecules Produced by Combined Static and Radiative Electric Fields
We show that combined electrostatic and radiative fields can greatly amplify
the directional properties, such as axis orientation and alignment, of
symmetric top molecules. In our computational study, we consider all four
symmetry combinations of the prolate and oblate inertia and polarizability
tensors, as well as the collinear and perpendicular (or tilted) geometries of
the two fields. In, respectively, the collinear or perpendicular fields, the
oblate or prolate polarizability interaction due to the radiative field forces
the permanent dipole into alignment with the static field. Two mechanisms are
found to be responsible for the amplification of the molecules' orientation,
which ensues once the static field is turned on: (a) permanent-dipole coupling
of the opposite-parity tunneling doublets created by the oblate polarizability
interaction in collinear static and radiative fields; (b) hybridization of the
opposite parity states via the polarizability interaction and their coupling by
the permanent dipole interaction to the collinear or perpendicular static
field. In perpendicular fields, the oblate polarizability interaction, along
with the loss of cylindrical symmetry, is found to preclude the wrong-way
orientation, causing all states to become high-field seeking with respect to
the static field. The adiabatic labels of the states in the tilted fields
depend on the adiabatic path taken through the parameter space comprised of the
permanent and induced-dipole interaction parameters and the tilt angle between
the two field vectors
Concatenation of convolutional and block codes Final report
Comparison of concatenated and sequential decoding systems and convolutional code structural propertie
The power of sum-of-squares for detecting hidden structures
We study planted problems---finding hidden structures in random noisy
inputs---through the lens of the sum-of-squares semidefinite programming
hierarchy (SoS). This family of powerful semidefinite programs has recently
yielded many new algorithms for planted problems, often achieving the best
known polynomial-time guarantees in terms of accuracy of recovered solutions
and robustness to noise. One theme in recent work is the design of spectral
algorithms which match the guarantees of SoS algorithms for planted problems.
Classical spectral algorithms are often unable to accomplish this: the twist in
these new spectral algorithms is the use of spectral structure of matrices
whose entries are low-degree polynomials of the input variables. We prove that
for a wide class of planted problems, including refuting random constraint
satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community
detection in stochastic block models, planted clique, and others, eigenvalues
of degree-d matrix polynomials are as powerful as SoS semidefinite programs of
roughly degree d. For such problems it is therefore always possible to match
the guarantees of SoS without solving a large semidefinite program. Using
related ideas on SoS algorithms and low-degree matrix polynomials (and inspired
by recent work on SoS and the planted clique problem by Barak et al.), we prove
new nearly-tight SoS lower bounds for the tensor and sparse principal component
analysis problems. Our lower bounds for sparse principal component analysis are
the first to suggest that going beyond existing algorithms for this problem may
require sub-exponential time
Soft clustering analysis of galaxy morphologies: A worked example with SDSS
Context: The huge and still rapidly growing amount of galaxies in modern sky
surveys raises the need of an automated and objective classification method.
Unsupervised learning algorithms are of particular interest, since they
discover classes automatically. Aims: We briefly discuss the pitfalls of
oversimplified classification methods and outline an alternative approach
called "clustering analysis". Methods: We categorise different classification
methods according to their capabilities. Based on this categorisation, we
present a probabilistic classification algorithm that automatically detects the
optimal classes preferred by the data. We explore the reliability of this
algorithm in systematic tests. Using a small sample of bright galaxies from the
SDSS, we demonstrate the performance of this algorithm in practice. We are able
to disentangle the problems of classification and parametrisation of galaxy
morphologies in this case. Results: We give physical arguments that a
probabilistic classification scheme is necessary. The algorithm we present
produces reasonable morphological classes and object-to-class assignments
without any prior assumptions. Conclusions: There are sophisticated automated
classification algorithms that meet all necessary requirements, but a lot of
work is still needed on the interpretation of the results.Comment: 18 pages, 19 figures, 2 tables, submitted to A
Chern-Simons Theory and Topological Strings
We review the relation between Chern-Simons gauge theory and topological
string theory on noncompact Calabi-Yau spaces. This relation has made possible
to give an exact solution of topological string theory on these spaces to all
orders in the string coupling constant. We focus on the construction of this
solution, which is encoded in the topological vertex, and we emphasize the
implications of the physics of string/gauge theory duality for knot theory and
for the geometry of Calabi-Yau manifolds.Comment: 46 pages, RMP style, 25 figures, minor corrections, references adde
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