Floating point computing has dominated scientific computations since the 1950s because of its versatile ability to approximate real numbers with both sufficiently high precision and sufficiently great range. Since 1985, two specific formats have become standard across the industry, one with 32 bits which is adequate for many purposes and one with 64 bits which allows numbers with both much higher precision and much greater range for applications where these are needed. However, during the past decade, deep learning has come to dominate computation, and narrower floating point formats with lower precision and reduced range have steadily become more common. In this article we discuss this phenomenon, why it is happening, how various 16-bit, 8-bit and even 4-bit floating point formats have been developed, and efforts to standardize these formats.
We propose a new mathematical formalism in which all kinds of tensors and of operations on those tensors are represented as simple function compositions. Tensors are functions from indices to values. An element-wise operation (e.g. an activation function) is a composition with a function applied to values; re-indexing (e.g. reshape, transposition) is a composition with a function applied to indices; and a reduction is a composition with both an index function, mapping each index to several indices, and a reduction operation (e.g. product, sum, max). Combined with tensor stacking, these compositions can represent most interesting kinds of tensors (e.g. dense, jagged, all kinds of sparsity patterns, matrix factorizations), as well as lazy versions of almost every operation available in modern tensor computation frameworks, including einsums between tensors of any kind. Identities between compositions can then be used to rewrite such expressions into equivalent ones, without evaluating them. For instance, the product of two sparse matrices can be rewritten so that only products between non-zero values are computed, turning a sparse problem into a dense one. Using this algebra as the intermediate representation of a tensor computation engine would shift the focus from defining operations on tensors (e.g. einsum between sparse tensors) to defining an algebra on index functions. This should reduce the codebase complexity from $\mathcal{O}(mn^2)$, with $m$ the number of binary operations between tensors and $n$ the number of tensor types, to $\mathcal{O}(m + k^2)$, with $k$ the number of types of index functions. A draft implementation of a tensor computation framework based on this algebra is publicly available.
We present a novel GPU-accelerated global solver for constrained nonlinear programming (NLP) inspired by the efficient linear bound propagation framework in neural network (NN) verification. Existing work on NN verification, a problem that can be cast as feasibility detection for a nonlinear objective, demonstrated orders-of-magnitude speedups with the linear bound propagation-based framework when compared to traditional solvers. Extending this paradigm to global nonlinear programming requires addressing many challenges: a global constrained NLP solver must effectively exploit all, possibly unstructured, nonlinear constraints to improve the objective, whereas NN verification typically handles simple input box constraints over a well-structured NN and does not directly optimize an objective function. We address these challenges by fully exploiting the linear bounds produced by bound propagation: these linear bounds can be viewed as relaxations of nonlinear constraints and used to provide lower bounds, check infeasibility, and guide the search for a primal solution. In addition, we propose tighter linear relaxations for bilinear, quotient, and other commonly used nonlinear functions in NLP problems, and a projected augmented Lagrangian method tightly coupled to our dual solving procedure to produce primal solutions. Our method solves 421 of 505 bounded and continuous NLP problems in the GAMS and MINLPLib benchmarks under a 180-second limit, more than each of the six global solvers we compare with, including Gurobi and BARON (commercial), and SCIP (open-source). Sixteen of these problems are certified by none of the six global solvers; all of them contain exponential, logarithmic, or trigonometric terms, which are often challenging for traditional approaches. Project webpage is at https://verified-intelligence.github.io/GEAR.
Accurate heart rate (HR) and interbeat interval (IBI) estimation from wearable electrocardiogram (ECG) is essential for continuous physiological monitoring, but motion artifacts can create false QRS-like peaks and obscure true R-peaks. Conventional detectors and local interval-repair rules process detections independently and may become unreliable when several consecutive beats are corrupted. We present ARTA (Artifact-Robust Temporal Alignment), a graph-based combinatorial framework that estimates the heartbeat sequence through constrained temporal optimization. ARTA combines candidates from multiple ECG detectors, represents them as nodes in a directed acyclic graph and connects candidates through physiologically plausible RR intervals. Candidate confidence uses detector agreement and local ECG quality, while history-dependent transition costs capture recent IBI consistency and a soft heart-rate prior. Beam-search dynamic programming selects a coherent low-cost path across the recording, suppressing false detections and recovering missed beats when plausible auxiliary candidates are available. On the IEEE Signal Processing Cup exercise dataset with electrode-motion artifacts from -2 to 12 dB, ARTA achieved an HR mean absolute error of 1.79 beats per minute, 54% lower than the strongest single-detector baseline, with 92.38% of windows within +/-5 beats per minute, an R-peak F1 score of 0.95, and an IBI mean absolute error of 6.52 ms. Cross-dataset evaluation on the European ST-T Database further assessed transfer to long clinical ambulatory recordings and a label-free reliability analysis examined conditional performance without using reference annotations for segment selection. ARTA is single-channel, training-free, interpretable and computationally lightweight, operating approximately 27 times faster than real time with peak memory below 2.1 MB in a single-thread CPU simulation.
We introduce dro, an open-source Python library for distributionally robust optimization (DRO) for regression and classification problems. The library implements 15 DRO formulations and 9 backbone models, enabling 85 distinct DRO methods. Furthermore, dro is compatible with both scikit-learn and PyTorch through a unified interface, and uses approximation techniques to reduce runtime for some computationally demanding formulations. Comprehensive documentation is available at https://python-dro.org.