Publications

This article, gathered and elaborated from a lecture by Eero Simoncelli at the 2024 Analytical Connectionism Summer School, reviews several approaches for modeling the probabilistic distribution of natural images and their interaction with the problem of image denoising. The lecture starts with the Gaussian spectral model of the 1950s as a conceptual foundation and quantitative baseline, followed by sparse coding models which took hold in the 1990s. These statistical models of natural images can be used as prior probability distributions for solving inverse problems such as denoising, using a Bayesian framework. Finally, the lecture describes recent work in machine learning in which the process of constructing a denoiser is reversed: a neural network is trained to solve the denoising problem without first specifying a prior distribution, and this trained network is subsequently used as an implicit model of the distribution of natural images. Images can be drawn from this implicit model through a reverse diffusion process, and the model can also be used to solve inference problems. This allows researchers to investigate the extent to which these DNNs are generalizing beyond their training data (as necessary for accurately modeling the distribution of natural images) as opposed to memorizing the images they were trained on.

We introduce a new arbitrarily high-order method for the rapid evaluation of hyperbolic potentials (space-time integrals involving the Green’s function for the scalar wave equation). With M points in the spatial discretization and Nt time steps of size Δt, a naive implementation would require O(M2Nt2) work in dimensions where the weak Huygens’ principle applies. We avoid this all-to-all interaction using a smoothly windowed decomposition into a local part, treated directly, plus a history part, approximated by a NF-term Fourier series. In one dimension, our method requires O((M+NFlogNF)Nt) work, with NF=O(1/Δt), by exploiting the non-uniform fast Fourier transform. We demonstrate the method’s performance for time-domain scattering problems involving a large number M of springs (point scatterers) attached to a vibrating string at arbitrary locations, with either periodic or free-space boundary conditions. We typically achieve 10-digit accuracy, and include tests for M up to a million.

While the thalamus is known to relay and modulate sensory signals to the cortex, whether it also participates in active computation and intrinsic signal generation remains unresolved. The anterodorsal nucleus of the thalamus broadcasts the head-direction (HD) signal, which is generated in the brainstem, particularly in the upstream lateral mammillary nucleus, and thalamic HD cells remain coordinated even during sleep. Here, by recording and manipulating neuronal activity along the mammillary–thalamic–cortical pathway, we show that coherence among thalamic HD cells persists even when their upstream inputs are decorrelated, particularly during non-Rapid Eye Movement sleep. These findings suggest that thalamic circuits are sufficient to generate and maintain coherent population dynamics in the absence of structured input.

We introduce the Information-Estimation Metric (IEM), a novel form of distance function derived from an underlying continuous probability density over a domain of signals. The IEM is rooted in a fundamental relationship between information theory and estimation theory, which links the log-probability of a signal with the errors of an optimal denoiser, applied to noisy observations of the signal. In particular, the IEM between a pair of signals is obtained by comparing their denoising error vectors over a range of noise amplitudes. Geometrically, this amounts to comparing the score vector fields of the blurred density around the signals over a range of blur levels. We prove that the IEM is a valid global metric and derive a closed-form expression for its local second-order approximation, which yields a Riemannian metric. For Gaussian-distributed signals, the IEM coincides with the Mahalanobis distance. But for more complex distributions, it adapts, both locally and globally, to the geometry of the distribution. In practice, the IEM can be computed using a learned denoiser (analogous to generative diffusion models) and solving a one-dimensional integral. To demonstrate the value of our framework, we learn an IEM on the ImageNet database. Experiments show that this IEM is competitive with or outperforms state-of-the-art supervised image quality metrics in predicting human perceptual judgments.

Chain-of-Thought (CoT) prompting significantly improves reasoning in Large Language Models, yet the temporal dynamics of the underlying representation geometry remain poorly understood. We investigate these dynamics by applying Manifold Capacity Theory (MCT) to a compositional Boolean logic task, allowing us to quantify the linear separability of latent representations without the confounding factors of probe training. Our analysis reveals that reasoning manifests as a transient geometric pulse, where concept manifolds are untangled into linearly separable subspaces immediately prior to computation and rapidly compressed thereafter. This behavior diverges from standard linear probe accuracy, which remains high long after computation, suggesting a fundamental distinction between information that is merely retrievable and information that is geometrically prepared for processing. We interpret this phenomenon as \emph{Dynamic Manifold Management}, a mechanism where the model dynamically modulates representational capacity to optimize the bandwidth of the residual stream throughout the reasoning chain.

Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial differential equation itself, one first evaluates a volume integral to account for the source distribution within the domain, followed by solving a boundary integral equation to impose the specified boundary conditions. Here, we present a new fast algorithm which is easy to implement and compatible with virtually any discretization technique, including unstructured domain triangulations, such as those used in standard finite element or finite volume methods. Our approach combines earlier work on potential theory for the heat equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT), and the dual-space multilevel kernel-splitting (DMK) framework. It is insensitive to flaws in the triangulation, permitting not just nonconforming elements, but arbitrary aspect ratio triangles, gaps and various other degeneracies. On a single CPU core, the scheme computes the solution at a rate comparable to that of the fast Fourier transform (FFT) in work per gridpoint.

While methods based on density-functional perturbation theory have dramatically improved our understanding of electron-phonon contributions to transport in materials, methods for accurately capturing electron-electron scattering relevant to low temperatures have seen significantly less development. The case of high-conductivity, moderately correlated materials characterized by low scattering rates is particularly challenging, since exquisite numerical precision of the low-energy electronic structure is required. Recent methodological advancements to density-functional theory combined with dynamical mean-field theory (DFT+DMFT), including adaptive Brillouin-zone integration and numerically precise self-energies, enable a rigorous investigation of electron-electron scattering in such materials. In particular, these tools may be leveraged to perform a robust scattering-rate analysis on both real- and imaginary-frequency axes. Applying this methodology to a subset of ABO$_3$ perovskite oxides -- SrVO$_3$, SrMoO$_3$, PbMoO$_3$, and SrRuO$_3$ -- we demonstrate its ability to qualitatively and quantitatively describe electron-electron contributions to the temperature-dependent direct-current resistivity. This combination of numerical techniques offers fundamental insight into the role of electronic correlations in transport phenomena and provides a predictive tool for identifying materials with potential for technological applications.

Spatial orientation enables animals to navigate their environment by rapidly mapping the external world and remembering key locations. In mammals, the head-direction (HD) system is an essential component of the navigation system of the brain. Although the tuning of neurons in other areas of this system is unstable—evidenced, for example, by the change in the spatial tuning of hippocampal place cells across days—the
stability of the neuronal code that underlies the sense of direction remains unclear. Here, by longitudinally tracking the activity of the same HD cells in the post-subiculum of freely moving mice, we show stability and plasticity at two levels. Although the population structure remained highly conserved across environments and over time, subtle shifts in population coherence encoded environment identity. In addition, the HD system established a distinct, environment-specific alignment between its internal representation and external landmarks, which persisted for weeks, even
after a single exposure. These findings suggest that the HD system forms long-lasting orientation memories that are anchored to specific environments.

We analyze, theoretically and empirically, the performance of generative diffusion models based on \emph{blind denoisers}, in which the denoiser is not given the noise amplitude in either the training or sampling processes. Assuming that the data distribution has low intrinsic dimensionality, we prove that blind denoising diffusion models (BDDMs), despite not having access to the noise amplitude, \emph{automatically} track a particular \emph{implicit} noise schedule along the reverse process. Our analysis shows that BDDMs can accurately sample from the data distribution in polynomially many steps as a function of the intrinsic dimension. Empirical results corroborate these mathematical findings on both synthetic and image data, demonstrating that the noise variance is accurately estimated from the noisy image. Remarkably, we observe that schedule-free BDDMs produce samples of higher quality compared to their non-blind counterparts. We provide evidence that this performance gain arises because BDDMs correct the mismatch between the true residual noise (of the image) and the noise assumed by the schedule used in non-blind diffusion models.

This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points (K,E) in the dispersion relation of tight-binding models accounting for macroscopic variations. We show that such wave packets are well approximated over long times by macroscopic models with varying orders of accuracy. Our main applications are in the analysis of single- and multilayer graphene tight-binding Hamiltonians modeling macroscopic variations such as those generated by shear or twist. Numerical simulations illustrate the theoretical findings.