278 research outputs found
Congruence properties of depths in some random trees
Consider a random recusive tree with n vertices. We show that the number of
vertices with even depth is asymptotically normal as n tends to infinty. The
same is true for the number of vertices of depth divisible by m for m=3, 4 or
5; in all four cases the variance grows linearly. On the other hand, for m at
least 7, the number is not asymptotically normal, and the variance grows faster
than linear in n. The case m=6 is intermediate: the number is asymptotically
normal but the variance is of order n log n.
This is a simple and striking example of a type of phase transition that has
been observed by other authors in several cases. We prove, and perhaps explain,
this non-intuitive behavious using a translation to a generalized Polya urn.
Similar results hold for a random binary search tree; now the number of
vertices of depth divisible by m is asymptotically normal for m at most 8 but
not for m at least 9, and the variance grows linearly in the first case both
faster in the second. (There is no intermediate case.)
In contrast, we show that for conditioned Galton-Watson trees, including
random labelled trees and random binary trees, there is no such phase
transition: the number is asymptotically normal for every m.Comment: 23 page
Recursive-Based PCG Methods for Toeplitz Systems with Nonnegative Generating Functions
In this paper, we consider the solutions of symmetric positive definite, but ill-conditioned, Toeplitz systems An x = b. Here we propose to solve the system by the recursive-based preconditioned conjugate gradient method. The idea is to use the inverse of Am (the principal submatrix of An with the Gohberg--Semencul formula as a preconditioner for An. The inverse of Am can be generated recursively by using the formula until m is small enough. The construction of the preconditioners requires only the entries of An and does not require the explicit knowledge of the generating function f of An. We show that if f is a nonnegative, bounded, and piecewise continuous even function with a finite number of zeros of even order, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges very quickly. Numerical results are included to illustrate the effectiveness of our approach.published_or_final_versio
A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition
In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for
the primary visual cortex of mammals. This model is neurophysiologically
justified. Further developments of this theory lead to efficient algorithms for
image reconstruction, based upon the consideration of an associated
hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or
certain of its improvements) is a left-invariant structure over the group
of rototranslations of the plane. Here, we propose a semi-discrete
version of this theory, leading to a left-invariant structure over the group
, restricting to a finite number of rotations. This apparently very
simple group is in fact quite atypical: it is maximally almost periodic, which
leads to much simpler harmonic analysis compared to Based upon this
semi-discrete model, we improve on previous image-reconstruction algorithms and
we develop a pattern-recognition theory that leads also to very efficient
algorithms in practice.Comment: 123 pages, revised versio
Large FHE Gates from tensored homomorphic accumulator
The main bottleneck of all known Fully Homomorphic Encryption schemes lies in the bootstrapping procedure invented by Gentry (STOC’09). The cost of this procedure can be mitigated either using Homomorphic SIMD techniques, or by performing larger computation per bootstrapping procedure.In this work, we propose new techniques allowing to perform more operations per bootstrapping in FHEW-type schemes (EUROCRYPT’13). While maintaining the quasi-quadratic Õ(n2) complexity of the whole cycle, our new scheme allows to evaluate gates with Ω(log n) input bits, which constitutes a quasi-linear speed-up. Our scheme is also very well adapted to large threshold gates, natively admitting up to Ω(n) inputs. This could be helpful for homomorphic evaluation of neural networks.Our theoretical contribution is backed by a preliminary prototype implementation, which can perform 6-to-6 bit gates in less than 10s on a single core, as well as threshold gates over 63 input bits even faster.<p
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