Andres Fernandez

Felix Dangel, Runa Eschenhagen, Weronika Ormaniec et al.

Structured large matrices are prevalent in machine learning. A particularly important class is curvature matrices like the Hessian, which are central to understanding the loss landscape of neural nets (NNs), and enable second-order optimization, uncertainty quantification, model pruning, data attribution, and more. However, curvature computations can be challenging due to the complexity of automatic differentiation, and the variety and structural assumptions of curvature proxies, like sparsity and Kronecker factorization. In this position paper, we argue that linear operators -- an interface for performing matrix-vector products -- provide a general, scalable, and user-friendly abstraction to handle curvature matrices. To support this position, we developed $\textit{curvlinops}$, a library that provides curvature matrices through a unified linear operator interface. We demonstrate with $\textit{curvlinops}$ how this interface can hide complexity, simplify applications, be extensible and interoperable with other libraries, and scale to large NNs.

Andres Fernandez, Frank Schneider, Maren Mahsereci et al.

Recently, it has been observed that when training a deep neural net with SGD, the majority of the loss landscape's curvature quickly concentrates in a tiny *top* eigenspace of the loss Hessian, which remains largely stable thereafter. Independently, it has been shown that successful magnitude pruning masks for deep neural nets emerge early in training and remain stable thereafter. In this work, we study these two phenomena jointly and show that they are connected: We develop a methodology to measure the similarity between arbitrary parameter masks and Hessian eigenspaces via Grassmannian metrics. We identify *overlap* as the most useful such metric due to its interpretability and stability. To compute *overlap*, we develop a matrix-free algorithm based on sketched SVDs that allows us to compute over 1000 Hessian eigenpairs for nets with over 10M parameters --an unprecedented scale by several orders of magnitude. Our experiments reveal an *overlap* between magnitude parameter masks and top Hessian eigenspaces consistently higher than chance-level, and that this effect gets accentuated for larger network sizes. This result indicates that *top Hessian eigenvectors tend to be concentrated around larger parameters*, or equivalently, that *larger parameters tend to align with directions of larger loss curvature*. Our work provides a methodology to approximate and analyze deep learning Hessians at scale, as well as a novel insight on the structure of their eigenspace.

Andres Fernandez, Felix Dangel, Philipp Hennig et al.

Many relevant machine learning and scientific computing tasks involve high-dimensional linear operators accessible only via costly matrix-vector products. In this context, recent advances in sketched methods have enabled the construction of *either* low-rank *or* diagonal approximations from few matrix-vector products. This provides great speedup and scalability, but approximation errors arise due to the assumed simpler structure. This work introduces SKETCHLORD, a method that simultaneously estimates both low-rank *and* diagonal components, targeting the broader class of Low-Rank *plus* Diagonal (LoRD) linear operators. We demonstrate theoretically and empirically that this joint estimation is superior also to any sequential variant (diagonal-then-low-rank or low-rank-then-diagonal). Then, we cast SKETCHLORD as a convex optimization problem, leading to a scalable algorithm. Comprehensive experiments on synthetic (approximate) LoRD matrices confirm SKETCHLORD's performance in accurately recovering these structures. This positions it as a valuable addition to the structured approximation toolkit, particularly when high-fidelity approximations are desired for large-scale operators, such as the deep learning Hessian.

Andres Fernandez, Juan Azcarreta, Cagdas Bilen et al.

Recent work in online speech spectrogram inversion effectively combines Deep Learning with the Gradient Theorem to predict phase derivatives directly from magnitudes. Then, phases are estimated from their derivatives via least squares, resulting in a high quality reconstruction. In this work, we introduce three innovations that drastically reduce computational cost, while maintaining high quality: Firstly, we introduce a novel neural network architecture with just 8k parameters, 30 times smaller than previous state of the art. Secondly, increasing latency by 1 hop size allows us to further halve the cost of the neural inference step. Thirdly, we we observe that the least squares problem features a tridiagonal matrix and propose a linear-complexity solver for the least squares step that leverages tridiagonality and positive-semidefiniteness, achieving a speedup of several orders of magnitude. We release samples online.

Andres Fernandez, Mark D. Plumbley

The goal of Unsupervised Anomaly Detection (UAD) is to detect anomalous signals under the condition that only non-anomalous (normal) data is available beforehand. In UAD under Domain-Shift Conditions (UAD-S), data is further exposed to contextual changes that are usually unknown beforehand. Motivated by the difficulties encountered in the UAD-S task presented at the 2021 edition of the Detection and Classification of Acoustic Scenes and Events (DCASE) challenge, we visually inspect Uniform Manifold Approximations and Projections (UMAPs) for log-STFT, log-mel and pretrained Look, Listen and Learn (L3) representations of the DCASE UAD-S dataset. In our exploratory investigation, we look for two qualities, Separability (SEP) and Discriminative Support (DSUP), and formulate several hypotheses that could facilitate diagnosis and developement of further representation and detection approaches. Particularly, we hypothesize that input length and pretraining may regulate a relevant tradeoff between SEP and DSUP. Our code as well as the resulting UMAPs and plots are publicly available.