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We investigate the fundamental optimization question of minimizing a target function $f$, whose gradients are expensive to compute or have limited availability, given access to some auxiliary side function $h$ whose gradients are cheap or more available. This formulation captures many settings of practical relevance, such as i) re-using batches in SGD, ii) transfer learning, iii) federated learning, iv) training with compressed models/dropout, etc. We propose two generic new algorithms that apply in all these settings and prove that we can benefit from this framework using only an assumption on the Hessian similarity between the target and side information. A benefit is obtained when this similarity measure is small, we also show a potential benefit from stochasticity when the auxiliary noise is correlated with that of the target function.

Distributed maximization of a submodular function in the MapReduce model has received much attention, culminating in two frameworks that allow a centralized algorithm to be run in the MR setting without loss of approximation, as long as the centralized algorithm satisfies a certain consistency property - which had only been shown to be satisfied by the standard greedy and continous greedy algorithms. A separate line of work has studied parallelizability of submodular maximization in the adaptive complexity model, where each thread may have access to the entire ground set. For the size-constrained maximization of a monotone and submodular function, we show that several sublinearly adaptive algorithms satisfy the consistency property required to work in the MR setting, which yields highly practical parallelizable and distributed algorithms. Also, we develop the first linear-time distributed algorithm for this problem with constant MR rounds. Finally, we provide a method to increase the maximum cardinality constraint for MR algorithms at the cost of additional MR rounds.

Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local PCA, a local version of principal component analysis. Our results accommodate for noisy non-uniform data distribution with the noise that may vary across the manifold, and allow simultaneous estimation at multiple points. Crucially, all of the constants appearing in our bound are explicitly described. The proof uses a matrix concentration inequality to estimate covariance matrices and a Wasserstein distance bound for quantifying nonlinearity of the underlying manifold and non-uniformity of the probability measure.

We can define the error distribution as the limiting distribution of the error between the solution $Y$ of a given stochastic differential equation (SDE) and its numerical approximation $\hat{Y}^{(m)}$, weighted by the convergence rate between the two. A goal when studying the error distribution is to provide a way of determination for error distributions for any SDE and numerical scheme that converge to the exact solution. By dividing the error into a main term and a remainder term in a particular way, the author shows that the remainder term can be negligible compared to the main term under certain suitable conditions. Under these conditions, deriving the error distribution reduces to deriving the limiting distribution of the main term. Even if the dimension is one, there are unsolved problems about the asymptotic behavior of the error when the SDE has a drift term and $0<H\leq 1/3$, but our result in the one-dimensional case can be adapted to any Hurst exponent. The main idea of the proof is to define a stochastic process $Y^{m, \rho}$ with the parameter $\rho$ interpolating between $Y$ and $\hat{Y}^{(m)}$ and to estimate the asymptotic expansion for it. Using this estimate, we determine the error distribution of the ($k$)-Milstein scheme and of the Crank-Nicholson scheme in unsolved cases.

Bilinear based models are powerful and widely used approaches for Knowledge Graphs Completion (KGC). Although bilinear based models have achieved significant advances, these studies mainly concentrate on posterior properties (based on evidence, e.g. symmetry pattern) while neglecting the prior properties. In this paper, we find a prior property named "the law of identity" that cannot be captured by bilinear based models, which hinders them from comprehensively modeling the characteristics of KGs. To address this issue, we introduce a solution called Unit Ball Bilinear Model (UniBi). This model not only achieves theoretical superiority but also offers enhanced interpretability and performance by minimizing ineffective learning through minimal constraints. Experiments demonstrate that UniBi models the prior property and verify its interpretability and performance.

An introductory exposition of the virtual element method (VEM) is provided. The intent is to make this method more accessible to those unfamiliar with VEM. Familiarity with the finite element method for solving 2D linear elasticity problems is assumed. Derivations relevant to successful implementation are covered. Some theory is covered, but the focus here is on implementation and results. Examples are given that illustrate the utility of the method. Numerical results are provided to help researchers implement and verify their own results.

Answering questions that require reading texts in an image is challenging for current models. One key difficulty of this task is that rare, polysemous, and ambiguous words frequently appear in images, e.g., names of places, products, and sports teams. To overcome this difficulty, only resorting to pre-trained word embedding models is far from enough. A desired model should utilize the rich information in multiple modalities of the image to help understand the meaning of scene texts, e.g., the prominent text on a bottle is most likely to be the brand. Following this idea, we propose a novel VQA approach, Multi-Modal Graph Neural Network (MM-GNN). It first represents an image as a graph consisting of three sub-graphs, depicting visual, semantic, and numeric modalities respectively. Then, we introduce three aggregators which guide the message passing from one graph to another to utilize the contexts in various modalities, so as to refine the features of nodes. The updated nodes have better features for the downstream question answering module. Experimental evaluations show that our MM-GNN represents the scene texts better and obviously facilitates the performances on two VQA tasks that require reading scene texts.

Incompleteness is a common problem for existing knowledge graphs (KGs), and the completion of KG which aims to predict links between entities is challenging. Most existing KG completion methods only consider the direct relation between nodes and ignore the relation paths which contain useful information for link prediction. Recently, a few methods take relation paths into consideration but pay less attention to the order of relations in paths which is important for reasoning. In addition, these path-based models always ignore nonlinear contributions of path features for link prediction. To solve these problems, we propose a novel KG completion method named OPTransE. Instead of embedding both entities of a relation into the same latent space as in previous methods, we project the head entity and the tail entity of each relation into different spaces to guarantee the order of relations in the path. Meanwhile, we adopt a pooling strategy to extract nonlinear and complex features of different paths to further improve the performance of link prediction. Experimental results on two benchmark datasets show that the proposed model OPTransE performs better than state-of-the-art methods.

Visual Question Answering (VQA) models have struggled with counting objects in natural images so far. We identify a fundamental problem due to soft attention in these models as a cause. To circumvent this problem, we propose a neural network component that allows robust counting from object proposals. Experiments on a toy task show the effectiveness of this component and we obtain state-of-the-art accuracy on the number category of the VQA v2 dataset without negatively affecting other categories, even outperforming ensemble models with our single model. On a difficult balanced pair metric, the component gives a substantial improvement in counting over a strong baseline by 6.6%.

Detecting carried objects is one of the requirements for developing systems to reason about activities involving people and objects. We present an approach to detect carried objects from a single video frame with a novel method that incorporates features from multiple scales. Initially, a foreground mask in a video frame is segmented into multi-scale superpixels. Then the human-like regions in the segmented area are identified by matching a set of extracted features from superpixels against learned features in a codebook. A carried object probability map is generated using the complement of the matching probabilities of superpixels to human-like regions and background information. A group of superpixels with high carried object probability and strong edge support is then merged to obtain the shape of the carried object. We applied our method to two challenging datasets, and results show that our method is competitive with or better than the state-of-the-art.