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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 \emph{exact} and \emph{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 $a_1,..., a_N$ each of whose coordinate has value bounded by one and bit-complexity $s$. * Approximate version: Given additionally an accuracy parameter $t$, the algorithm computes rational numbers $\beta_1,\ldots, \beta_N$ such that $|f(a_i) - \beta_i| \leq \frac{1}{2^t}$ for all $i$, and has a running time of $((Nm + d^m)(s + t))^{1 + o(1)}$ for all $m$ and all sufficiently large $d$. * Exact version (when over rationals): Given additionally a bound $c$ on the bit-complexity of all evaluations, the algorithm computes the rational numbers $f(a_1), ... , f(a_N)$, in time $((Nm + d^m)(s + c))^{1 + o(1)}$ for all $m$ and all sufficiently large $d$. . 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 (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 (FOCS 2021) and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar & Umans (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.

Image retrieval with natural language feedback offers the promise of catalog search based on fine-grained visual features that go beyond objects and binary attributes, facilitating real-world applications such as e-commerce. Our Modality-Agnostic Attention Fusion (MAAF) model combines image and text features and outperforms existing approaches on two visual search with modifying phrase datasets, Fashion IQ and CSS, and performs competitively on a dataset with only single-word modifications, Fashion200k. We also introduce two new challenging benchmarks adapted from Birds-to-Words and Spot-the-Diff, which provide new settings with rich language inputs, and we show that our approach without modification outperforms strong baselines. To better understand our model, we conduct detailed ablations on Fashion IQ and provide visualizations of the surprising phenomenon of words avoiding "attending" to the image region they refer to., Comment: 14 pages, 8 figures

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., Comment: 10 pages

In this paper, we propose learning an embedding function for content-based image retrieval within the e-commerce domain using the triplet loss and an online sampling method that constructs triplets from within a minibatch. We compare our method to several strong baselines as well as recent works on the DeepFashion and Stanford Online Product datasets. Our approach significantly outperforms the state-of-the-art on the DeepFashion dataset. With a modification to favor sampling minibatches from a single product category, the same approach demonstrates competitive results when compared to the state-of-the-art for the Stanford Online Products dataset.

We prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets $A,B$ of the finite field $\mathbb{F}_p$, the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset $A+B$ in terms of the sizes of the sets $A$ and $B$. Our theorem considers a general linear map $L: \mathbb{F}_p^n \to \mathbb{F}_p^m$, and subsets $A_1, \ldots, A_n \subseteq \mathbb{F}_p$, and gives a lower bound on the size of $L(A_1 \times A_2 \times \ldots \times A_n)$ in terms of the sizes of the sets $A_1, \ldots, A_n$. Our proof uses Alon's Combinatorial Nullstellensatz and a variation of the polynomial method., Comment: 16 pages, 0 figures

As an extension of Polya's classical result on random walks on the square grids ($\Z^d$), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after $n$ steps is at most $n^{-d/2 - d/(d-2) +o(1)}$, which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is $n^{-\omega (1)}$, which is much worse than higher dimensions. In dimension 3, we prove an upper bound of order $n^{-4 +o(1)}$. We discover a new conjecture concerning incidences between spheres and points in $\R^3$, which, if holds, would improve the bound to $n^{-9/2 +o(1)}$, which is consistent % with the $d \ge 4$ case. to the $d \ge 4$ case. This conjecture resembles Szemer\'edi-Trotter type results and is of independent interest.