28,854 research outputs found
Note on Integer Factoring Algorithms II
This note introduces a new class of integer factoring algorithms. Two
versions of this method will be described, deterministic and probabilistic.
These algorithms are practical, and can factor large classes of balanced
integers N = pq, p < q < 2p in superpolynomial time. Further, an extension of
the Fermat factoring method is proposed.Comment: 23 Page
Distributed Arithmetic Coding for the Asymmetric Slepian-Wolf problem
Distributed source coding schemes are typically based on the use of channels
codes as source codes. In this paper we propose a new paradigm, termed
"distributed arithmetic coding", which exploits the fact that arithmetic codes
are good source as well as channel codes. In particular, we propose a
distributed binary arithmetic coder for Slepian-Wolf coding with decoder side
information, along with a soft joint decoder. The proposed scheme provides
several advantages over existing Slepian-Wolf coders, especially its good
performance at small block lengths, and the ability to incorporate arbitrary
source models in the encoding process, e.g. context-based statistical models.
We have compared the performance of distributed arithmetic coding with turbo
codes and low-density parity-check codes, and found that the proposed approach
has very competitive performance.Comment: submitted to IEEE Transactions on Signal processing, Nov. 2007.
Revised version accepted with minor revision
Carries, group theory, and additive combinatorics
Given a group G and a normal subgroup H we study the problem of choosing
coset representatives with few carries
Consensus Driven by the Geometric Mean
Consensus networks are usually understood as arithmetic mean driven dynamical
averaging systems. In applications, however, network dynamics often describe
inherently non-arithmetic and non-linear consensus processes. In this paper, we
propose and study three novel consensus protocols driven by geometric mean
averaging: a polynomial, an entropic, and a scaling-invariant protocol, where
terminology characterizes the particular non-linearity appearing in the
respective differential protocol equation. We prove exponential convergence to
consensus for positive initial conditions. For the novel protocols we highlight
connections to applied network problems: The polynomial consensus system is
structured like a system of chemical kinetics on a graph. The entropic
consensus system converges to the weighted geometric mean of the initial
condition, which is an immediate extension of the (weighted) average consensus
problem. We find that all three protocols generate gradient flows of free
energy on the simplex of constant mass distribution vectors albeit in different
metrics. On this basis, we propose a novel variational characterization of the
geometric mean as the solution of a non-linear constrained optimization problem
involving free energy as cost function. We illustrate our findings in numerical
simulations
Off-diagonal ordered Ramsey numbers of matchings
For ordered graphs and , the ordered Ramsey number is the
smallest such that every red/blue edge coloring of the complete graph on
vertices contains either a blue copy of or a red copy of
, where the embedding must preserve the relative order of vertices. One
number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the
"off-diagonal" ordered Ramsey number , where is an ordered
matching on vertices. In particular, Conlon et al. asked what asymptotic
bounds (in ) can be obtained for , where the maximum is
over all ordered matchings on vertices. The best-known upper bound is
, whereas the best-known lower bound is , and Conlon et al. hypothesize that for every ordered matching . We resolve two special cases
of this conjecture. We show that the off-diagonal ordered Ramsey numbers for
matchings in which edges do not cross are nearly linear. We also prove a truly
sub-quadratic upper bound for random matchings with interval chromatic number
.Comment: 15 pages, 3 figure
A note on the random greedy independent set algorithm
Let r be a fixed constant and let H be an r-uniform, D-regular hypergraph on
N vertices. Assume further that D > N^\epsilon for some \epsilon>0. Consider
the random greedy algorithm for forming an independent set in H. An independent
set is chosen at random by iteratively choosing vertices at random to be in the
independent set. At each step we choose a vertex uniformly at random from the
collection of vertices that could be added to the independent set (i.e. the
collection of vertices v with the property that v is not in the current
independent set I and I \cup {v} contains no edge of H). Note that this process
terminates at a maximal subset of vertices with the property that this set
contains no edge of H; that is, the process terminates at a maximal independent
set. We prove that if H satisfies certain degree and codegree conditions then
there are \Omega(N ((log N) / D)^{1/(r-1)}) vertices in the independent set
produced by the random greedy algorithm with high probability. This result
generalizes a lower bound on the number of steps in the H-free process due to
Bohman and Keevash and produces objects of interest in additive combinatorics.Comment: 24 page
MergeShuffle: A Very Fast, Parallel Random Permutation Algorithm
This article introduces an algorithm, MergeShuffle, which is an extremely
efficient algorithm to generate random permutations (or to randomly permute an
existing array). It is easy to implement, runs in time, is
in-place, uses random bits, and can be parallelized
accross any number of processes, in a shared-memory PRAM model. Finally, our
preliminary simulations using OpenMP suggest it is more efficient than the
Rao-Sandelius algorithm, one of the fastest existing random permutation
algorithms.
We also show how it is possible to further reduce the number of random bits
consumed, by introducing a second algorithm BalancedShuffle, a variant of the
Rao-Sandelius algorithm which is more conservative in the way it recursively
partitions arrays to be shuffled. While this algorithm is of lesser practical
interest, we believe it may be of theoretical value.
Our full code is available at: https://github.com/axel-bacher/mergeshuffleComment: Preliminary draft. 12 pages, 1 figure, 3 algorithms, implementation
code at https://github.com/axel-bacher/mergeshuffl
On The Discrepancy of Quasi-progressions
The 2-colouring discrepancy of arithmetic progressions is a well-known
problem in combinatorial discrepancy theory. In 1964, Roth proved that if each
integer from 0 to N is coloured red or blue, there is some arithmetic
progression in which the number of reds and the number of blues differ by at
least (1/20) N^{1/4}. In 1996, Matousek and Spencer showed that this estimate
is sharp up to a constant. The analogous question for homogeneous arithmetic
progressions (i.e., the ones containing 0) was raised by Erdos in the 1930s,
and it is still not known whether the discrepancy is unbounded. However, it is
easy to construct partial colourings with density arbitrarily close to 1 such
that all homogeneous arithmetic progressions have bounded discrepancy.
A related problem concerns the discrepancy of quasi-progressions. A
quasi-progression consists of successive multiples of a real number, with each
multiple rounded down to the nearest integer. In 1986, Beck showed that given
any 2-colouring, the quasi-progressions corresponding to almost all real
numbers in (1, \infty) have discrepancy at least log* N, the inverse of the
tower function. We improve the lower bound to (log N)^{1/4 - o(1)}, and also
show that there is some quasi-progression with discrepancy at least (1/50)
N^{1/6}. Our results remain valid even if the 2-colouring is replaced by a
partial colouring of positive density.Comment: 15 page
Implicit Regularization in Deep Learning May Not Be Explainable by Norms
Mathematically characterizing the implicit regularization induced by
gradient-based optimization is a longstanding pursuit in the theory of deep
learning. A widespread hope is that a characterization based on minimization of
norms may apply, and a standard test-bed for studying this prospect is matrix
factorization (matrix completion via linear neural networks). It is an open
question whether norms can explain the implicit regularization in matrix
factorization. The current paper resolves this open question in the negative,
by proving that there exist natural matrix factorization problems on which the
implicit regularization drives all norms (and quasi-norms) towards infinity.
Our results suggest that, rather than perceiving the implicit regularization
via norms, a potentially more useful interpretation is minimization of rank. We
demonstrate empirically that this interpretation extends to a certain class of
non-linear neural networks, and hypothesize that it may be key to explaining
generalization in deep learning
Fast Approximate Computations with Cauchy Matrices and Polynomials
Multipoint polynomial evaluation and interpolation are fundamental for modern
symbolic and numerical computing. The known algorithms solve both problems over
any field of constants in nearly linear arithmetic time, but the cost grows to
quadratic for numerical solution. We fix this discrepancy: our new numerical
algorithms run in nearly linear arithmetic time. At first we restate our goals
as the multiplication of an n-by-n Vandermonde matrix by a vector and the
solution of a Vandermonde linear system of n equations. Then we transform the
matrix into a Cauchy structured matrix with some special features. By
exploiting them, we approximate the matrix by a generalized hierarchically
semiseparable matrix, which is a structured matrix of a different class.
Finally we accelerate our solution to the original problems by applying Fast
Multipole Method to the latter matrix. Our resulting numerical algorithms run
in nearly optimal arithmetic time when they perform the above fundamental
computations with polynomials, Vandermonde matrices, transposed Vandermonde
matrices, and a large class of Cauchy and Cauchy-like matrices. Some of our
techniques may be of independent interest.Comment: 31 pages, 7 figures, 9 table
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