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• , University of Chicago (read with Lek-Heng Lim in Quanta Magazine) 

Noncommutative positivstellensatz and stochastic gradient descent (Friday 1:15pm)

Abstract: Most students with even a fleeting brush with algebraic geometry would have seen Hilbert’s Nullstellensatz over the complex field. In mainstream algebraic geometry, it is a starting point for defining increasingly abstract geometric objects over progressively esoteric fields and rings, ultimately taking it to dizzying heights. But at least from an applied math perspective, the answers to the most natural questions about Nullstellensatz — what if we do it over reals? what if we have inequalities? what if we have variables that don’t commute? what if we have functions more general than polynomials? — cannot be found in mainstream algebraic geometry. Somewhat surprisingly, these have been answered by applied mathematicians working on control engineering problems. We will discuss one of these extensions of Nullstellensatz and see how it applies to solve an open problem about stochastic gradient descent. This is based on joint work with Zehua Lai.

Attention is a smoothed cubic spline (Saturday 9:00am)

Abstract: We highlight a perhaps important but hitherto unobserved insight: The attention module in a transformer is a smoothed cubic spline. Viewed in this manner, this mysterious but critical component of a transformer becomes a natural development of an old notion deeply entrenched in classical approximation theory. More precisely, we show that with ReLU-activation, attention, masked attention, encoder-decoder attention are all cubic splines. As every component in a transformer is constructed out of compositions of various attention modules (= cubic splines) and feed forward neural networks (= linear splines), all its components — encoder, decoder, and encoder-decoder blocks; multilayered encoders and decoders; the transformer itself — are cubic or higher-order splines. If we assume the Pierce-Birkhoff conjecture, then the converse also holds, i.e., every spline is a ReLU-activated encoder. Since a spline is generally just C^2, one way to obtain a smoothed C^\infty-version is by replacing
ReLU with a smooth activation; and if this activation is chosen to be SoftMax, we recover the original transformer as proposed by Vaswani et al. This insight sheds light on the nature of the transformer by casting it entirely in terms of splines, one of the best known and thoroughly understood objects in applied mathematics. This is joint work with Zehua Lai (Texas) and Yucong Liu (Georgia Tech).

• , Ohio University

Wow! Did you just defeat the Curse of Dimensionality? (Friday 4:00pm)

Abstract: Many machine learning (and other) tasks need to represent functions of many variables. The straightforward representation using a grid requires the number of points to grow exponentially with respect to the dimension, an effect known as the curse of dimensionality. Much effort has gone into developing methods to cope with or defeat the curse. If you can defeat the curse (without cheating), then I will give you a dollar.

What the Sine-of-a-Sum example tells us about Sums of Separable Functions and Canonical Tensor Decompositions (Saturday 11:05am)

Abstract: Many machine learning (and other) tasks reduce to finding a way to represent a function of many variables without triggering the curse of dimensionality. Representations using sums of separable functions, which are the continuous version of canonical tensor approximations, provide one approach. Applying this approach to the simple example of the sine of the sum of the variables has revealed an unreasonable number of insights into the approach. Along the way it uncovered a nonstandard trigonometric identity.