Quantized Compressive K-Means

The recent framework of compressive statistical learning aims at designing tractable learning algorithms that use only a heavily compressed representation-or sketch-of massive datasets. Compressive K-Means (CKM) is such a method: it estimates the centroids of data clusters from pooled, non-linear, random signatures of the learning examples. While this approach significantly reduces computational time on very large datasets, its digital implementation wastes acquisition resources because the learning examples are compressed only after the sensing stage. The present work generalizes the sketching procedure initially defined in Compressive K-Means to a large class of periodic nonlinearities including hardware-friendly implementations that compressively acquire entire datasets. This idea is exemplified in a Quantized Compressive K-Means procedure, a variant of CKM that leverages 1-bit universal quantization (i.e. retaining the least significant bit of a standard uniform quantizer) as the periodic sketch nonlinearity. Trading for this resource-efficient signature (standard in most acquisition schemes) has almost no impact on the clustering performances, as illustrated by numerical experiments

Ward type identities for the 2d Anderson model at weak disorder

Using the particular momentum conservation laws in dimension d=2, we can rewrite the Anderson model in terms of low momentum long range fields, at the price of introducing electron loops. The corresponding loops satisfy a Ward type identity, hence are much smaller than expected. This fact should be useful for a study of the weak-coupling model in the middle of the spectrum of the free Hamiltonian.Comment: LaTeX 2e document using AMS symbols, 25 pages and 32 eps figure

Renormalization of the 2-point function of the Hubbard model at half-filling

We prove that the two dimensional Hubbard model at finite temperature T and half-filling is analytic in the coupling constant in a radius at least $c/(\log T)^2$. We also study the self-energy through a new two-particle irreducible expansion and prove that this model is not a Fermi liquid, but a Luttinger liquid with logarithmic corrections. The techniques used are borrowed from constructive field theory so the result is mathematically rigorous and completely non-perturbative. Together with earlier results on the existence of two dimensional Fermi liquids, this new result proves that the nature of interacting Fermi systems in two dimensions depends on the shape of the Fermi surface.Comment: 45 pages, 28 figure

The Hubbard model at half-filling, part III: the lower bound on the self-energy

We complete the proof that the two-dimensional Hubbard model at half-filling is not a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing a lower bound on a second derivative in momentum of the first non-trivial self-energy graph.Comment: 31 pages, 4 figure

Scaling behaviour of three-dimensional group field theory

Group field theory is a generalization of matrix models, with triangulated pseudomanifolds as Feynman diagrams and state sum invariants as Feynman amplitudes. In this paper, we consider Boulatov's three-dimensional model and its Freidel-Louapre positive regularization (hereafter the BFL model) with a ?ultraviolet' cutoff, and study rigorously their scaling behavior in the large cutoff limit. We prove an optimal bound on large order Feynman amplitudes, which shows that the BFL model is perturbatively more divergent than the former. We then upgrade this result to the constructive level, using, in a self-contained way, the modern tools of constructive field theory: we construct the Borel sum of the BFL perturbative series via a convergent ?cactus' expansion, and establish the ?ultraviolet' scaling of its Borel radius. Our method shows how the ?sum over trian- gulations' in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory