Georgia Institute of Technology College of Sciences

The brain performs efficient, adaptable, and robust computations of noisy sensory information in changing environments. While artificial neural networks have achieved remarkable successes in recent years, the brain's computational capacity is yet to be matched. To understand mechanisms underlying the exquisite computational efficiency and flexibility of the brain, complex architecture and dynamics of the biological neural networks should be studied. In this talk, I will give a broad overview of recent research projects from my group, that investigate links between neural coding and network structures using data-driven modeling.

Multimodal contrastive learning is a methodology for linking different data modalities, such as images and text. It is typically framed as the identification of a set of encoders—one for each modality—that align representations within a common latent space. In this presentation, we interpret contrastive learning as the optimization of encoders that define conditional probability distributions, for each modality conditioned on the other, in a way consistent with the available data. This probabilistic perspective suggests two natural generalizations of contrastive learning: (i) the introduction of novel probabilistic loss functions, and (ii) the use of alternative metrics for measuring alignment in the common latent space. We investigate these generalizations of the classical approach in the multivariate Gaussian setting by viewing latent space identification as a low-rank matrix approximation problem. The proposed framework is further studied through numerical experiments on multivariate Gaussians, the labeled MNIST dataset, and a data assimilation application in oceanography.

This talk is about an application of combinatorial algebraic geometry to complex/arithmetic dynamics. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic ramification point. Per_n is an affine algebraic curve, defined over Q, parametrizing degree-2 rational maps with an n-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is G_n is irreducible over Q? (2) Is Per_n connected? We show that if G_n is irreducible over Q, then Per_n is irreducible over C, and is therefore connected. In order to do this, we find a Q-rational smooth point on a projective completion of Per_n — this Q-rational smooth point represents a special degeneration of degree-2 self-maps.

The classical Wiener-Wintner Theorem says that for all measure preserving systems, and bounded functions f, there is a set of full measure so that the averages below converge for all continuous functions  g from the circle (R/Z)  to the complex numbers.

N^{-1} \sum_{n=1}^N  g( \pi n) f(T^n). 

We extend this result to averages over the prime integers. The proof uses structure of measure preserving systems, higher order Fourier analysis, and the Heath-Brown approximate to the von Mangoldt function.  A key result is a surprisingly small  Gowers norm estimate for the Heath-Brown approximate with fixed height.  

Joint work with  Y. Chen, A. Fragkos,  J. Fornal, B. Krause, and H. Mousavi.