arXiv subjects

Albert Gu

Publications and source records attributed to Albert Gu.

50 records · Page 3Linked to original sources

arXiv2021

Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers

Recurrent neural networks (RNNs), temporal convolutions, and neural differential equations (NDEs) are popular families of deep learning models for time-series data, each with unique strengths and tradeoffs in modeling power and computational efficiency. We introduce a simple sequence model inspired by control systems that generalizes these approaches while addressing their shortcomings. The Linear State-Space Layer (LSSL) maps a sequence $u \mapsto y$ by simply simulating a linear continuous-time state-space representation $\dot{x} = Ax + Bu, y = Cx + Du$. Theoretically, we show that LSSL models are closely related to the three aforementioned families of models and inherit their strengths. For example, they generalize convolutions to continuous-time, explain common RNN heuristics, and share features of NDEs such as time-scale adaptation. We then incorporate and generalize recent theory on continuous-time memorization to introduce a trainable subset of structured matrices $A$ that endow LSSLs with long-range memory. Empirically, stacking LSSL layers into a simple deep neural network obtains state-of-the-art results across time series benchmarks for long dependencies in sequential image classification, real-world healthcare regression tasks, and speech. On a difficult speech classification task with length-16000 sequences, LSSL outperforms prior approaches by 24 accuracy points, and even outperforms baselines that use hand-crafted features on 100x shorter sequences.

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arXiv2021

HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections

This paper studies Principal Component Analysis (PCA) for data lying in hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of subspaces spanned by these directions, (2) a method of projection onto subspaces that preserves information in these directions, and (3) an objective to optimize, namely the variance explained by projections. We generalize each of these concepts to the hyperbolic space and propose HoroPCA, a method for hyperbolic dimensionality reduction. By focusing on the core problem of extracting principal directions, HoroPCA theoretically better preserves information in the original data such as distances, compared to previous generalizations of PCA. Empirically, we validate that HoroPCA outperforms existing dimensionality reduction methods, significantly reducing error in distance preservation. As a data whitening method, it improves downstream classification by up to 3.9% compared to methods that don't use whitening. Finally, we show that HoroPCA can be used to visualize hyperbolic data in two dimensions.

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arXiv2021

Kaleidoscope: An Efficient, Learnable Representation For All Structured Linear Maps

Modern neural network architectures use structured linear transformations, such as low-rank matrices, sparse matrices, permutations, and the Fourier transform, to improve inference speed and reduce memory usage compared to general linear maps. However, choosing which of the myriad structured transformations to use (and its associated parameterization) is a laborious task that requires trading off speed, space, and accuracy. We consider a different approach: we introduce a family of matrices called kaleidoscope matrices (K-matrices) that provably capture any structured matrix with near-optimal space (parameter) and time (arithmetic operation) complexity. We empirically validate that K-matrices can be automatically learned within end-to-end pipelines to replace hand-crafted procedures, in order to improve model quality. For example, replacing channel shuffles in ShuffleNet improves classification accuracy on ImageNet by up to 5%. K-matrices can also simplify hand-engineered pipelines -- we replace filter bank feature computation in speech data preprocessing with a learnable kaleidoscope layer, resulting in only 0.4% loss in accuracy on the TIMIT speech recognition task. In addition, K-matrices can capture latent structure in models: for a challenging permuted image classification task, a K-matrix based representation of permutations is able to learn the right latent structure and improves accuracy of a downstream convolutional model by over 9%. We provide a practically efficient implementation of our approach, and use K-matrices in a Transformer network to attain 36% faster end-to-end inference speed on a language translation task.

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arXiv2020

Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations

Fast linear transforms are ubiquitous in machine learning, including the discrete Fourier transform, discrete cosine transform, and other structured transformations such as convolutions. All of these transforms can be represented by dense matrix-vector multiplication, yet each has a specialized and highly efficient (subquadratic) algorithm. We ask to what extent hand-crafting these algorithms and implementations is necessary, what structural priors they encode, and how much knowledge is required to automatically learn a fast algorithm for a provided structured transform. Motivated by a characterization of fast matrix-vector multiplication as products of sparse matrices, we introduce a parameterization of divide-and-conquer methods that is capable of representing a large class of transforms. This generic formulation can automatically learn an efficient algorithm for many important transforms; for example, it recovers the $O(N \log N)$ Cooley-Tukey FFT algorithm to machine precision, for dimensions $N$ up to $1024$. Furthermore, our method can be incorporated as a lightweight replacement of generic matrices in machine learning pipelines to learn efficient and compressible transformations. On a standard task of compressing a single hidden-layer network, our method exceeds the classification accuracy of unconstrained matrices on CIFAR-10 by 3.9 points -- the first time a structured approach has done so -- with 4X faster inference speed and 40X fewer parameters.

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arXiv2020

HiPPO: Recurrent Memory with Optimal Polynomial Projections

A central problem in learning from sequential data is representing cumulative history in an incremental fashion as more data is processed. We introduce a general framework (HiPPO) for the online compression of continuous signals and discrete time series by projection onto polynomial bases. Given a measure that specifies the importance of each time step in the past, HiPPO produces an optimal solution to a natural online function approximation problem. As special cases, our framework yields a short derivation of the recent Legendre Memory Unit (LMU) from first principles, and generalizes the ubiquitous gating mechanism of recurrent neural networks such as GRUs. This formal framework yields a new memory update mechanism (HiPPO-LegS) that scales through time to remember all history, avoiding priors on the timescale. HiPPO-LegS enjoys the theoretical benefits of timescale robustness, fast updates, and bounded gradients. By incorporating the memory dynamics into recurrent neural networks, HiPPO RNNs can empirically capture complex temporal dependencies. On the benchmark permuted MNIST dataset, HiPPO-LegS sets a new state-of-the-art accuracy of 98.3%. Finally, on a novel trajectory classification task testing robustness to out-of-distribution timescales and missing data, HiPPO-LegS outperforms RNN and neural ODE baselines by 25-40% accuracy.

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arXiv2020

From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering

Similarity-based Hierarchical Clustering (HC) is a classical unsupervised machine learning algorithm that has traditionally been solved with heuristic algorithms like Average-Linkage. Recently, Dasgupta reframed HC as a discrete optimization problem by introducing a global cost function measuring the quality of a given tree. In this work, we provide the first continuous relaxation of Dasgupta's discrete optimization problem with provable quality guarantees. The key idea of our method, HypHC, is showing a direct correspondence from discrete trees to continuous representations (via the hyperbolic embeddings of their leaf nodes) and back (via a decoding algorithm that maps leaf embeddings to a dendrogram), allowing us to search the space of discrete binary trees with continuous optimization. Building on analogies between trees and hyperbolic space, we derive a continuous analogue for the notion of lowest common ancestor, which leads to a continuous relaxation of Dasgupta's discrete objective. We can show that after decoding, the global minimizer of our continuous relaxation yields a discrete tree with a (1 + epsilon)-factor approximation for Dasgupta's optimal tree, where epsilon can be made arbitrarily small and controls optimization challenges. We experimentally evaluate HypHC on a variety of HC benchmarks and find that even approximate solutions found with gradient descent have superior clustering quality than agglomerative heuristics or other gradient based algorithms. Finally, we highlight the flexibility of HypHC using end-to-end training in a downstream classification task.

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arXiv2020

Model Patching: Closing the Subgroup Performance Gap with Data Augmentation

Classifiers in machine learning are often brittle when deployed. Particularly concerning are models with inconsistent performance on specific subgroups of a class, e.g., exhibiting disparities in skin cancer classification in the presence or absence of a spurious bandage. To mitigate these performance differences, we introduce model patching, a two-stage framework for improving robustness that encourages the model to be invariant to subgroup differences, and focus on class information shared by subgroups. Model patching first models subgroup features within a class and learns semantic transformations between them, and then trains a classifier with data augmentations that deliberately manipulate subgroup features. We instantiate model patching with CAMEL, which (1) uses a CycleGAN to learn the intra-class, inter-subgroup augmentations, and (2) balances subgroup performance using a theoretically-motivated subgroup consistency regularizer, accompanied by a new robust objective. We demonstrate CAMEL's effectiveness on 3 benchmark datasets, with reductions in robust error of up to 33% relative to the best baseline. Lastly, CAMEL successfully patches a model that fails due to spurious features on a real-world skin cancer dataset.

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arXiv2020

Improving the Gating Mechanism of Recurrent Neural Networks

Gating mechanisms are widely used in neural network models, where they allow gradients to backpropagate more easily through depth or time. However, their saturation property introduces problems of its own. For example, in recurrent models these gates need to have outputs near 1 to propagate information over long time-delays, which requires them to operate in their saturation regime and hinders gradient-based learning of the gate mechanism. We address this problem by deriving two synergistic modifications to the standard gating mechanism that are easy to implement, introduce no additional hyperparameters, and improve learnability of the gates when they are close to saturation. We show how these changes are related to and improve on alternative recently proposed gating mechanisms such as chrono initialization and Ordered Neurons. Empirically, our simple gating mechanisms robustly improve the performance of recurrent models on a range of applications, including synthetic memorization tasks, sequential image classification, language modeling, and reinforcement learning, particularly when long-term dependencies are involved.

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arXiv2019

Sparse Recovery for Orthogonal Polynomial Transforms

In this paper we consider the following sparse recovery problem. We have query access to a vector $\vx \in \R^N$ such that $\vhx = \vF \vx$ is $k$-sparse (or nearly $k$-sparse) for some orthogonal transform $\vF$. The goal is to output an approximation (in an $\ell_2$ sense) to $\vhx$ in sublinear time. This problem has been well-studied in the special case that $\vF$ is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time $O(k \cdot \mathrm{polylog} N)$. However, for transforms $\vF$ other than the DFT (or closely related transforms like the Discrete Cosine Transform), the question is much less settled. In this paper we give sublinear-time algorithms---running in time $\poly(k \log(N))$---for solving the sparse recovery problem for orthogonal transforms $\vF$ that arise from orthogonal polynomials. More precisely, our algorithm works for any $\vF$ that is an orthogonal polynomial transform derived from Jacobi polynomials. The Jacobi polynomials are a large class of classical orthogonal polynomials (and include Chebyshev and Legendre polynomials as special cases), and show up extensively in applications like numerical analysis and signal processing. One caveat of our work is that we require an assumption on the sparsity structure of the sparse vector, although we note that vectors with random support have this property with high probability. Our approach is to give a very general reduction from the $k$-sparse sparse recovery problem to the $1$-sparse sparse recovery problem that holds for any flat orthogonal polynomial transform; then we solve this one-sparse recovery problem for transforms derived from Jacobi polynomials.

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arXiv2019

A Kernel Theory of Modern Data Augmentation

Data augmentation, a technique in which a training set is expanded with class-preserving transformations, is ubiquitous in modern machine learning pipelines. In this paper, we seek to establish a theoretical framework for understanding data augmentation. We approach this from two directions: First, we provide a general model of augmentation as a Markov process, and show that kernels appear naturally with respect to this model, even when we do not employ kernel classification. Next, we analyze more directly the effect of augmentation on kernel classifiers, showing that data augmentation can be approximated by first-order feature averaging and second-order variance regularization components. These frameworks both serve to illustrate the ways in which data augmentation affects the downstream learning model, and the resulting analyses provide novel connections between prior work in invariant kernels, tangent propagation, and robust optimization. Finally, we provide several proof-of-concept applications showing that our theory can be useful for accelerating machine learning workflows, such as reducing the amount of computation needed to train using augmented data, and predicting the utility of a transformation prior to training.

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arXiv2019

Learning Compressed Transforms with Low Displacement Rank

The low displacement rank (LDR) framework for structured matrices represents a matrix through two displacement operators and a low-rank residual. Existing use of LDR matrices in deep learning has applied fixed displacement operators encoding forms of shift invariance akin to convolutions. We introduce a class of LDR matrices with more general displacement operators, and explicitly learn over both the operators and the low-rank component. This class generalizes several previous constructions while preserving compression and efficient computation. We prove bounds on the VC dimension of multi-layer neural networks with structured weight matrices and show empirically that our compact parameterization can reduce the sample complexity of learning. When replacing weight layers in fully-connected, convolutional, and recurrent neural networks for image classification and language modeling tasks, our new classes exceed the accuracy of existing compression approaches, and on some tasks also outperform general unstructured layers while using more than 20x fewer parameters.

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arXiv2018

Representation Tradeoffs for Hyperbolic Embeddings

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

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arXiv2017

A Two Pronged Progress in Structured Dense Matrix Multiplication

Matrix-vector multiplication is one of the most fundamental computing primitives. Given a matrix $A\in\mathbb{F}^{N\times N}$ and a vector $b$, it is known that in the worst case $Θ(N^2)$ operations over $\mathbb{F}$ are needed to compute $Ab$. A broad question is to identify classes of structured dense matrices that can be represented with $O(N)$ parameters, and for which matrix-vector multiplication can be performed sub-quadratically. One such class of structured matrices is the orthogonal polynomial transforms, whose rows correspond to a family of orthogonal polynomials. Other well known classes include the Toeplitz, Hankel, Vandermonde, Cauchy matrices and their extensions that are all special cases of a ldisplacement rank property. In this paper, we make progress on two fronts: 1. We introduce the notion of recurrence width of matrices. For matrices with constant recurrence width, we design algorithms to compute $Ab$ and $A^Tb$ with a near-linear number of operations. This notion of width is finer than all the above classes of structured matrices and thus we can compute multiplication for all of them using the same core algorithm. 2. We additionally adapt this algorithm to an algorithm for a much more general class of matrices with displacement structure: those with low displacement rank with respect to quasiseparable matrices. This class includes Toeplitz-plus-Hankel-like matrices, Discrete Cosine/Sine Transforms, and more, and captures all previously known matrices with displacement structure that we are aware of under a unified parametrization and algorithm. Our work unifies, generalizes, and simplifies existing state-of-the-art results in structured matrix-vector multiplication. Finally, we show how applications in areas such as multipoint evaluations of multivariate polynomials can be reduced to problems involving low recurrence width matrices.

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arXiv2013

The Power of Deferral: Maintaining a Constant-Competitive Steiner Tree Online

In the online Steiner tree problem, a sequence of points is revealed one-by-one: when a point arrives, we only have time to add a single edge connecting this point to the previous ones, and we want to minimize the total length of edges added. For two decades, we know that the greedy algorithm maintains a tree whose cost is O(log n) times the Steiner tree cost, and this is best possible. But suppose, in addition to the new edge we add, we can change a single edge from the previous set of edges: can we do much better? Can we maintain a tree that is constant-competitive? We answer this question in the affirmative. We give a primal-dual algorithm, and a novel dual-based analysis, that makes only a single swap per step (in addition to adding the edge connecting the new point to the previous ones), and such that the tree's cost is only a constant times the optimal cost. Previous results for this problem gave an algorithm that performed an amortized constant number of swaps: for each n, the number of swaps in the first n steps was O(n). We also give a simpler tight analysis for this amortized case.

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