74 research outputs found
Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems
We consider chains of random constraint satisfaction models that are
spatially coupled across a finite window along the chain direction. We
investigate their phase diagram at zero temperature using the survey
propagation formalism and the interpolation method. We prove that the SAT-UNSAT
phase transition threshold of an infinite chain is identical to the one of the
individual standard model, and is therefore not affected by spatial coupling.
We compute the survey propagation complexity using population dynamics as well
as large degree approximations, and determine the survey propagation threshold.
We find that a clustering phase survives coupling. However, as one increases
the range of the coupling window, the survey propagation threshold increases
and saturates towards the phase transition threshold. We also briefly discuss
other aspects of the problem. Namely, the condensation threshold is not
affected by coupling, but the dynamic threshold displays saturation towards the
condensation one. All these features may provide a new avenue for obtaining
better provable algorithmic lower bounds on phase transition thresholds of the
individual standard model
The Full Rank Condition for Sparse Random Matrices
We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. Inspired by low-density parity check codes, the family of random matrices that we investigate is very general and encompasses both matrices over finite fields and {0,1}-matrices over the rationals. The proof combines statistical physics-inspired coupling techniques with local limit arguments
Linear Programming Relaxations for Goldreich's Generators over Non-Binary Alphabets
Goldreich suggested candidates of one-way functions and pseudorandom
generators included in . It is known that randomly generated
Goldreich's generator using -wise independent predicates with input
variables and output variables is not pseudorandom generator with
high probability for sufficiently large constant . Most of the previous
works assume that the alphabet is binary and use techniques available only for
the binary alphabet. In this paper, we deal with non-binary generalization of
Goldreich's generator and derives the tight threshold for linear programming
relaxation attack using local marginal polytope for randomly generated
Goldreich's generators. We assume that input
variables are known. In that case, we show that when , there is an
exact threshold
such
that for , the LP relaxation can determine
linearly many input variables of Goldreich's generator if
, and that the LP relaxation cannot determine
input variables of Goldreich's generator if
. This paper uses characterization of LP solutions by
combinatorial structures called stopping sets on a bipartite graph, which is
related to a simple algorithm called peeling algorithm.Comment: 14 pages, 1 figur
The satisfiability threshold for random linear equations
Let be a random matrix over the finite field with
precisely non-zero entries per row and let be a random vector
chosen independently of . We identify the threshold up to which the
linear system has a solution with high probability and analyse the
geometry of the set of solutions. In the special case , known as the
random -XORSAT problem, the threshold was determined by [Dubois and Mandler
2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof
technique was subsequently extended to the cases [Falke and Goerdt
2012]. But the argument depends on technically demanding second moment
calculations that do not generalise to . Here we approach the problem from
the viewpoint of a decoding task, which leads to a transparent combinatorial
proof
Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems
We consider chains of random constraint satisfaction models that are spatially coupled across a finite window along the chain direction. We investigate their phase diagram at zero temperature using the survey propagation formalism and the interpolation method. We prove that the SAT-UNSAT phase transition threshold of an infinite chain is identical to the one of the individual standard model, and is therefore not affected by spatial coupling. We compute the survey propagation complexity using population dynamics as well as large degree approximations, and determine the survey propagation threshold. We find that a clustering phase survives coupling. However, as one increases the range of the coupling window, the survey propagation threshold increases and saturates towards the phase transition threshold. We also briefly discuss other aspects of the problem. Namely, the condensation threshold is not affected by coupling, but the dynamic threshold displays saturation towards the condensation one. All these features may provide a new avenue for obtaining better provable algorithmic lower bounds on phase transition thresholds of the individual standard mode
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Closest-vector problem and the zero-temperature p-spin landscape for lossy compression
We consider a high-dimensional random constrained optimization problem in which a set of binary variables is subjected to a linear system of equations. The cost function is a simple linear cost, measuring the Hamming distance with respect to a reference configuration. Despite its apparent simplicity, this problem exhibits a rich phenomenology. We show that different situations arise depending on the random ensemble of linear systems. When each variable is involved in at most two linear constraints, we show that the problem can be partially solved analytically, in particular we show that upon convergence, the zero-temperature limit of the cavity equations returns the optimal solution. We then study the geometrical properties of more general random ensembles. In particular we observe a range in the density of constraints at which the system enters a glassy phase where the cost function has many minima. Interestingly, the algorithmic performances are only sensitive to another phase transition affecting the structure of configurations allowed by the linear constraints. We also extend our results to variables belonging to GF(q), the Galois field of order q. We show that increasing the value of q allows to achieve a better optimum, which is confirmed by the replica-symmetric cavity method predictions
- …