A crucial feature of neural networks with a fixed network architecture is that they form a manifold by virtue of their continuously tunable weights, which underlies their ability to be trained by gradient descent. However, this conception of the space of neural networks is inadequate for understanding the computational processes the networks perform. For example, two neural networks trained to perform the same task may have vastly different weights, and yet implement the same high-level algorithms and computational processes (Maheswaranathan et al., 2019b). In this paper, we construct an algorithm which provides alternative parametrizations of the space of RNNs and CNNs with the goal of endowing a geometric structure that is more compatible with the high-level computational processes performed by neural networks. In particular, given a set of neural networks with the same or possibly different architectures (and possibly trained on different tasks), we find a parametrization of a low-dimensional submanifold of neural networks which approximately interpolates between these chosen "base models", as well as extrapolates beyond them. We can use such model embedding spaces to cluster neural networks and even compute model averages of neural networks. A key feature is that two points in model embedding space are nearby if they correspond to neural networks which implement similar high-level computational processes, in a manner to be described later. In this way, two neural networks may correspond to nearby points in model embedding space even if those neural networks have distinct weights or even architectures.

Self-training is a standard approach to semi-supervised learning where the learner's own predictions on unlabeled data are used as supervision during training. In this paper, we reinterpret this label assignment process as an optimal transportation problem between examples and classes, wherein the cost of assigning an example to a class is mediated by the current predictions of the classifier. This formulation facilitates a practical annealing strategy for label assignment and allows for the inclusion of prior knowledge on class proportions via flexible upper bound constraints. The solutions to these assignment problems can be efficiently approximated using Sinkhorn iteration, thus enabling their use in the inner loop of standard stochastic optimization algorithms. We demonstrate the effectiveness of our algorithm on the CIFAR-10, CIFAR-100, and SVHN datasets in comparison with FixMatch, a state-of-the-art self-training algorithm. Additionally, we elucidate connections between our proposed algorithm and existing confidence thresholded self-training approaches in the context of homotopy methods in optimization. Our code is available at https://github.com/stanford-futuredata/sinkhorn-label-allocation.

How can prior knowledge on the transformation invariances of a domain be incorporated into the architecture of a neural network? We propose Equivariant Transformers (ETs), a family of differentiable image-to-image mappings that improve the robustness of models towards pre-defined continuous transformation groups. Through the use of specially-derived canonical coordinate systems, ETs incorporate functions that are equivariant by construction with respect to these transformations. We show empirically that ETs can be flexibly composed to improve model robustness towards more complicated transformation groups in several parameters. On a real-world image classification task, ETs improve the sample efficiency of ResNet classifiers, achieving relative improvements in error rate of up to 15% in the limited data regime while increasing model parameter count by less than 1%.

We introduce a new sub-linear space data structure---the Weight-Median Sketch---that captures the most heavily weighted features in linear classifiers trained over data streams. This enables memory-limited execution of several statistical analyses over streams, including online feature selection, streaming data explanation, relative deltoid detection, and streaming estimation of pointwise mutual information. In contrast with related sketches that capture the most commonly occurring features (or items) in a data stream, the Weight-Median Sketch captures the features that are most discriminative of one stream (or class) compared to another. The Weight-Median sketch adopts the core data structure used in the Count-Sketch, but, instead of sketching counts, it captures sketched gradient updates to the model parameters. We provide a theoretical analysis of this approach that establishes recovery guarantees in the online learning setting, and demonstrate substantial empirical improvements in accuracy-memory trade-offs over alternatives, including count-based sketches and feature hashing.

We propose a simple and efficient approach to learning sparse models. Our approach consists of (1) projecting the data into a lower dimensional space, (2) learning a dense model in the lower dimensional space, and then (3) recovering the sparse model in the original space via compressive sensing. We apply this approach to Non-negative Matrix Factorization (NMF), tensor decomposition and linear classification---showing that it obtains $10\times$ compression with negligible loss in accuracy on real data, and obtains up to $5\times$ speedups. Our main theoretical contribution is to show the following result for NMF: if the original factors are sparse, then their projections are the sparsest solutions to the projected NMF problem. This explains why our method works for NMF and shows an interesting new property of random projections: they can preserve the solutions of non-convex optimization problems such as NMF.