Global $\mathcal{L}^2$ minimization with certainty via geometrically adapted gradient descent in Deep Learning

We consider the gradient descent flow widely used for the minimization of the $\mathcal{L}^2$ cost function in Deep Learning networks, and introduce two modified versions; one adapted for the overparametrized setting, and the other for the underparametrized setting. Both have a clear and natural invariant geometric meaning, taking into account the pullback vector bundle structure in the overparametrized, and the pushforward vector bundle structure in the underparametrized setting. In the overparametrized case, we prove that, provided that a rank condition holds, all orbits of the modified gradient descent drive the $\mathcal{L}^2$ cost to its global minimum at a uniform exponential convergence rate. We point out relations of the latter to sub-Riemannian geometry.

Non-approximability of constructive global $\mathcal{L}^2$ minimizers by gradient descent in Deep Learning

We analyze geometric aspects of the gradient descent algorithm in Deep Learning (DL) networks. In particular, we prove that the globally minimizing weights and biases for the $\mathcal{L}^2$ cost obtained constructively in [Chen-Munoz Ewald 2023] for underparametrized ReLU DL networks can generically not be approximated via the gradient descent flow. We therefore conclude that the method introduced in [Chen-Munoz Ewald 2023] is disjoint from the gradient descent method.

Geometric structure of Deep Learning networks and construction of global ${\mathcal L}^2$ minimizers

In this paper, we provide a geometric interpretation of the structure of Deep Learning (DL) networks, characterized by $L$ hidden layers, a ramp activation function, an ${\mathcal L}^2$ Schatten class (or Hilbert-Schmidt) cost function, and input and output spaces ${\mathbb R}^Q$ with equal dimension $Q\geq1$. The hidden layers are defined on spaces ${\mathbb R}^{Q}$, as well. We apply our recent results on shallow neural networks to construct an explicit family of minimizers for the global minimum of the cost function in the case $L\geq Q$, which we show to be degenerate. In the context presented here, the hidden layers of the DL network "curate" the training inputs by recursive application of a truncation map that minimizes the noise to signal ratio of the training inputs. Moreover, we determine a set of $2^Q-1$ distinct degenerate local minima of the cost function.

Geometric structure of shallow neural networks and constructive ${\mathcal L}^2$ cost minimization

In this paper, we provide a geometric interpretation of the structure of shallow neural networks characterized by one hidden layer, a ramp activation function, an ${\mathcal L}^2$ Schatten class (or Hilbert-Schmidt) cost function, input space ${\mathbb R}^M$, output space ${\mathbb R}^Q$ with $Q\leq M$, and training input sample size $N>QM$. We prove an upper bound on the minimum of the cost function of order $O(\delta_P$ where $\delta_P$ measures the signal to noise ratio of training inputs. We obtain an approximate optimizer using projections adapted to the averages $\overline{x_{0,j}}$ of training input vectors belonging to the same output vector $y_j$, $j=1,\dots,Q$. In the special case $M=Q$, we explicitly determine an exact degenerate local minimum of the cost function; the sharp value differs from the upper bound obtained for $Q\leq M$ by a relative error $O(\delta_P^2)$. The proof of the upper bound yields a constructively trained network; we show that it metrizes the $Q$-dimensional subspace in the input space ${\mathbb R}^M$ spanned by $\overline{x_{0,j}}$, $j=1,\dots,Q$. We comment on the characterization of the global minimum of the cost function in the given context.

Hierarchical Summarization for Longform Spoken Dialog

Every day we are surrounded by spoken dialog. This medium delivers rich diverse streams of information auditorily; however, systematically understanding dialog can often be non-trivial. Despite the pervasiveness of spoken dialog, automated speech understanding and quality information extraction remains markedly poor, especially when compared to written prose. Furthermore, compared to understanding text, auditory communication poses many additional challenges such as speaker disfluencies, informal prose styles, and lack of structure. These concerns all demonstrate the need for a distinctly speech tailored interactive system to help users understand and navigate the spoken language domain. While individual automatic speech recognition (ASR) and text summarization methods already exist, they are imperfect technologies; neither consider user purpose and intent nor address spoken language induced complications. Consequently, we design a two stage ASR and text summarization pipeline and propose a set of semantic segmentation and merging algorithms to resolve these speech modeling challenges. Our system enables users to easily browse and navigate content as well as recover from errors in these underlying technologies. Finally, we present an evaluation of the system which highlights user preference for hierarchical summarization as a tool to quickly skim audio and identify content of interest to the user.