Publications by Nicolas Macris

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We theoretically investigate the phenomena of generalization and memorization in diffusion models. Empirical studies suggest that these phenomena are influenced by model complexity and the size of the training dataset. In our experiments, we further observe that the number of noise samples per data sample (m) used during Denoising Score Matching (DSM) plays a significant and non-trivial role. We capture these behaviors and shed insights into their mechanisms by deriving asymptotically precise expressions fo

Investigations on spatially coupled codes have lead to the conjecture that, in the infinite size limit, the average input-output conditional entropy for spatially coupled low-density parity-check ensembles, over binary memoryless symmetric channels, equals ...

This paper is about a novel technique called spatial coupling and its application in the analysis of random constraint satisfaction problems (CSP). Spatial Coupling was recently invented in the area of error correcting codes where it has resulted in efficient capacity-achieving codes for a wide range of channels. However, this technique is not limited to problems in communications. It can be applied in the much broader context of graphical models. We describe here a general methodology for applying spatial

We consider a single-band approximation to the random Schr6dinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the entire spectrum of this Hamiltonian when the magnetic field is sufficiently high. We show that the whole spectrum is pure point, the energy coinciding with the first Landau level in the absence of a random potential being infinitely degenerate, while the ei

The simplest statistical-mechanical model of crystalline formation (or alloy formation) that includes electronic degrees of freedom is solved exactly in the limit of large spatial dimensions and infinite interaction strength. The solutions contain both second-order phase transitions and first-order phase transitions (that involve phase separation or segregation) which are likely to illustrate the precursor physics behind the static charge-stripe ordering in cuprate systems. In addition, we find that the spi

We study a new encoding scheme for lossy source compression based on spatially coupled low-density generatormatrix codes. We develop a belief-propagation guided-decimation algorithm, and show that this algorithm allows to approach the optimal distortion of ...

We consider estimating a matrix from noisy observations coming from an arbitrary additive bi-rotational invariant perturbation. We propose an estimator, which we conjecture is optimal among the class of rectangular rotational invariant estimators and can b ...

The subject of this paper is transmission over a general class of binary-input memoryless symmetric channels using error correcting codes based on sparse graphs, namely, low-density generator-matrix and low-density parity-check codes. The optimal (or ideal ...

We rigorously derive a single-letter variational expression for the mutual information of the asymmetric two-groups stochastic block model in the dense graph regime. Existing proofs in the literature are indirect, as they involve mapping the model to a ran ...

Spatial coupling has recently emerged as a powerful paradigm to construct graphical models that work well under low-complexity message-passing algorithms. Although much progress has been made on the analysis of spatially coupled models under message passin ...

The ground states of the one-dimensional Falicov-Kimball model are investigated in the small-coupling limit, using nearly degenerate perturbation theory. For rational electron and ion densities, respectively, equal to p/q, pi/q, with p relatively prime to q and pi/q close enough to 1/2, we find that in the ground state the ion configuration has a period q. The situation is analogous to the Peierls instability, where the usual arguments predict a period-q state that produces a gap at the Fermi level and is i

We consider an electron in two dimensions submitted to a magnetic field and to the potential of impurities. We show that when the electron is confined to a half-space by a planar wall described by a smooth increasing potential, the total Hamiltonian necessarily has a continuous spectrum in some intervals in between the Landau levels provided that both the amplitude and spatial variation of the impurity potential are sufficiently weak. The spatial decay of the impurity potential is not needed. In particular,

The aim of this paper is to show that spatial coupling can be viewed not only as a means to build better graphical models, but also as a tool to better understand uncoupled models. The starting point is the observation that some asymptotic properties of gr ...

It is shown that a correlation inequality of statistical mechanics can be applied to linear low-density parity-check codes. Thanks to this tool we prove that, under a natural assumption, the exponential growth rate of regular low-density parity-check (LDPC ...

Consider transmission over a binary additive white gaussian noise channel using a fixed low-density parity check code. We consider the posterior measure over the code bits and the corresponding correlation between two codebits, averaged over the noise real ...

We present a simplification of Lieb's proof of the flux phase conjecture for interacting fermion systems--such as the Hubbard model--at half-filling on a general class of graphs. The main ingredient is a procedure which transforms a class of fermionic Hamiltonians into reflection-positive form. The method can also be applied to other problems, which we briefly illustrate with two examples concerning the t-V model and an extended Falicov-Kimball model.

We consider the problem of estimating a rank-one nonsymmetric matrix under additive white Gaussian noise. The matrix to estimate can be written as the outer product of two vectors and we look at the special case in which both vectors are uniformly distribu ...

We consider a single-band approximation to the random Schro¨dinger operator in an external magnetic field. The random potential is taken to be constant on unit squares and i.i.d. on each square with a bounded distribution. We prove that the eigenstates corresponding to energies at the edges of the Landau band are localized. This is an important ingredient in the theory of the Quantum Hall Effect.

We investigate an encoding scheme for lossy compression based on spatially coupled Low-Density Generator-Matrix codes. The degree distributions are regular, or are Poisson on the code-bit side and check-regular which allows use for any compression rate. Th ...

We consider communication over the binary erasure and the binary additive white gaussian noise channels using fixed linear block codes and also appropriate ensembles of such codes. We show concentration of the magnetization over the channel realizations an ...

Nicolas Macris

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Denoising Score Matching with Random Features: Insights on Diffusion Models From Precise Learning Curves

How to Prove the Maxwell Conjecture Via Spatial Coupling - A Proof of Concept

New Bounds for Random Constraint Satisfaction Problems via Spatial Coupling

The nature of the spectrum for a Landau hamiltonian with delta impurities

Phase separation and the segregation principle in the infinite-U spinless Falicov-Kimball model

Lossy source coding via spatially coupled LDGM ensembles

Rectangular rotational invariant estimator for high-rank matrix estimation

Decay of correlations for sparse graph error correcting codes

Mutual Information for the Stochastic Block Model by the Adaptive Interpolation Method

Displacement Convexity - A Useful Framework for the Study of Spatially Coupled Codes

Phase separation in the binary-alloy problem: The one dimensional spinless Falicov Kimball model

On edge states in semi-infinite quantum Hall systems

Spatial Coupling as a Proof Technique and Three Applications

Griffiths Kelly Sherman correlation inequalities: a useful tool in the theory of error correcting codes

Decay of Correlations in Low Density Parity Check Codes: Low Noise Regime

On the flux phase conjecture at half filling: an improved proof

High-dimensional rank-one nonsymmetric matrix decomposition: the spherical case

Localisation in single Landau bands

Approaching the Rate-Distortion Limit by spatial Coupling with Belief Propagation and Decimation

Concentration of Magnetization for Linear Block Codes