We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both exact and approximate versions of the algorithm. The input to the algorithms are (1) coefficients of an m-variate polynomial f with degree d in each variable, and (2) points $\mathbf{a}_{1}, \ldots, \mathbf{a}_{N}$ each of whose coordinate has absolute value bounded by one. Approximate version: Given additionally an accuracy parameter t, the algorithm computes rational numbers $\beta_{1}, \ldots, \beta_{N}$ such that $\left|f\left(\mathbf{a}_{i}\right)-\beta_{i}\right| \leq 1 / 2^{t}$ for all i, and has a running time of $\left(\left(N m+d^{m}\right) t\right)^{1+o(1)}$ for all m and all sufficiently large d. Exact version (when over rationals): Given additionally a bound s on the bit-complexity of all the rational numbers in the input and output, the algorithm computes the rational numbers $f\left(\mathbf{a}_{1}\right), \ldots, f\left(\mathbf{a}_{N}\right)$, in time $\left(\left(N m+d^{m}\right) s\right)^{1+o(1)}$ for all m and all sufficiently large d. Our results also naturally extend to the case when the input is over the field of real or complex numbers under an appropriate standard model of representation of field elements in such fields.Prior to this work, a nearly-linear time algorithm for multivariate multipoint evaluation (exact or approximate) over any infinite field appears to be known only for the case of univariate polynomials, and was discovered in a recent work of Moroz [Proc. 62nd FOCS, 2021]. In this work, we extend this result from the univariate to the multivariate setting. However, our algorithm is based on ideas that seem to be conceptually different from those of Moroz [Proc. 62nd FOCS, 2021] and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar & Umans [Proc. 63rd FOCS, 2022] for multivariate multipoint evaluation over finite fields, and known efficient algorithms for the problems of rational number reconstruction and fast Chinese remaindering in computational number theory.

In this work, we develop a theoretical and experimental framework to study the effect of margin penalties on angular softmax losses, which have led to state-of-the-art performance in face recognition. We also introduce a new multiplicative margin which performs comparably to previously proposed additive margins when the model is trained to convergence. A regime of the margin parameters can lead to degenerate minima, but these can be reliably avoided through the use of two regularization techniques that we propose. Our theory predicts the minimal angular distance between sample embeddings and the correct and wrong class prototype vectors learned during training, and it suggests a new method to identify optimal margin parameters without expensive tuning. Finally, we conduct a thorough ablation study of the margin parameters in our proposed framework, and we characterize the sensitivity of generalization to each parameter both theoretically and through experiments on standard face recognition benchmarks.

Image captioning models typically follow an encoder-decoder architecture which uses abstract image feature vectors as input to the encoder. One of the most successful algorithms uses feature vectors extracted from the region proposals obtained from an object detector. In this work we introduce the Object Relation Transformer, that builds upon this approach by explicitly incorporating information about the spatial relationship between input detected objects through geometric attention. Quantitative and qualitative results demonstrate the importance of such geometric attention for image captioning, leading to improvements on all common captioning metrics on the MS-COCO dataset. Code is available at https://github.com/yahoo/objectrelationtransformer .