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Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method. Code is available at //github.com/HJ-harry/DDS

Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ gradient computations to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, which is therefore optimal in terms of sample complexity.

Triply periodic minimal surfaces (TPMSs) play a vital role in the design of porous structures, with applications in bone tissue engineering, chemical engineering, and the creation of lightweight models. However, fabrication of TPMSs via additive manufacturing is feasible only within a specific range of relative densities, termed the effective relative density range (EDR), outside of which TPMSs exhibit unmanufacturable features. In this study, the persistent homology is applied to theoretically calculate and extend the EDRs of TPMSs. The TPMSs with extended EDRs are referred to as extended TPMSs. To achieve this, TPMSs are converted into implicit B-spline representation through fitting. By analyzing the symmetry of TPMSs, a partial fitting method is utilized to preserve the symmetry and enhance fitting precision. A topological objective function is modeled based on the understanding of topological features, resulting in extended TPMSs that possess extended EDRs while maintaining a high degree of similarity to the original TPMSs. Experimental validation confirms the effectiveness of the approach in extending the EDRs of TPMSs. Furthermore, the extended TPMSs demonstrate superior performance in porous model design and topology optimization compared to their original counterparts. The extended TPMSs with increased EDRs hold promise for replacing traditional TPMSs in applications that require porous structures with varying densities.

This paper leverages the framework of algorithms-with-predictions to design data structures for two fundamental dynamic graph problems: incremental topological ordering and cycle detection. In these problems, the input is a directed graph on $n$ nodes, and the $m$ edges arrive one by one. The data structure must maintain a topological ordering of the vertices at all times and detect if the newly inserted edge creates a cycle. The theoretically best worst-case algorithms for these problems have high update cost (polynomial in $n$ and $m$). In practice, greedy heuristics (that recompute the solution from scratch each time) perform well but can have high update cost in the worst case. In this paper, we bridge this gap by leveraging predictions to design a learned new data structure for the problems. Our data structure guarantees consistency, robustness, and smoothness with respect to predictions -- that is, it has the best possible running time under perfect predictions, never performs worse than the best-known worst-case methods, and its running time degrades smoothly with the prediction error. Moreover, we demonstrate empirically that predictions, learned from a very small training dataset, are sufficient to provide significant speed-ups on real datasets.

The task of precisely learning the probability distribution of rows within tabular data and producing authentic synthetic samples is both crucial and non-trivial. Wasserstein generative adversarial network (WGAN) marks a notable improvement in generative modeling, addressing the challenges faced by its predecessor, generative adversarial network. However, due to the mixed data types and multimodalities prevalent in tabular data, the delicate equilibrium between the generator and discriminator, as well as the inherent instability of Wasserstein distance in high dimensions, WGAN often fails to produce high-fidelity samples. To this end, we propose POTNet (Penalized Optimal Transport Network), a generative deep neural network based on a novel, robust, and interpretable marginally-penalized Wasserstein (MPW) loss. POTNet can effectively model tabular data containing both categorical and continuous features. Moreover, it offers the flexibility to condition on a subset of features. We provide theoretical justifications for the motivation behind the MPW loss. We also empirically demonstrate the effectiveness of our proposed method on four different benchmarks across a variety of real-world and simulated datasets. Our proposed model achieves orders of magnitude speedup during the sampling stage compared to state-of-the-art generative models for tabular data, thereby enabling efficient large-scale synthetic data generation.

Gradient methods are experiencing a growth in methodological and theoretical developments owing to the challenges of optimization problems arising in data science. Focusing on data science applications with expensive objective function evaluations yet inexpensive gradient function evaluations, gradient methods that never make objective function evaluations are either being rejuvenated or actively developed. However, as we show, such gradient methods are all susceptible to catastrophic divergence under realistic conditions for data science applications. In light of this, gradient methods which make use of objective function evaluations become more appealing, yet, as we show, can result in an exponential increase in objective evaluations between accepted iterates. As a result, existing gradient methods are poorly suited to the needs of optimization problems arising from data science. In this work, we address this gap by developing a generic methodology that economically uses objective function evaluations in a problem-driven manner to prevent catastrophic divergence and avoid an explosion in objective evaluations between accepted iterates. Our methodology allows for specific procedures that can make use of specific step size selection methodologies or search direction strategies, and we develop a novel step size selection methodology that is well-suited to data science applications. We show that a procedure resulting from our methodology is highly competitive with standard optimization methods on CUTEst test problems. We then show a procedure resulting from our methodology is highly favorable relative to standard optimization methods on optimization problems arising in our target data science applications. Thus, we provide a novel gradient methodology that is better suited to optimization problems arising in data science.

The problem of goal-oriented semantic filtering and timely source coding in multiuser communication systems is considered here. We study a distributed monitoring system in which multiple information sources, each observing a physical process, provide status update packets to multiple monitors having heterogeneous goals. Two semantic filtering schemes are first proposed as a means to admit or drop arrival packets based on their goal-dependent importance, which is a function of the intrinsic and extrinsic attributes of information and the probability of occurrence of each realization. Admitted packets at each sensor are then encoded and transmitted over block-fading wireless channels so that served monitors can timely fulfill their goals. A truncated error control scheme is derived, which allows transmitters to drop or retransmit undelivered packets based on their significance. Then, we formulate the timely source encoding optimization problem and analytically derive the optimal codeword lengths assigned to the admitted packets which maximize a weighted sum of semantic utility functions for all pairs of communicating sensors and monitors. Our analytical and numerical results provide the optimal design parameters for different arrival rates and highlight the improvement in timely status update delivery using the proposed semantic filtering, source coding, and error control schemes.

The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.

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.

We introduce a generic framework that reduces the computational cost of object detection while retaining accuracy for scenarios where objects with varied sizes appear in high resolution images. Detection progresses in a coarse-to-fine manner, first on a down-sampled version of the image and then on a sequence of higher resolution regions identified as likely to improve the detection accuracy. Built upon reinforcement learning, our approach consists of a model (R-net) that uses coarse detection results to predict the potential accuracy gain for analyzing a region at a higher resolution and another model (Q-net) that sequentially selects regions to zoom in. Experiments on the Caltech Pedestrians dataset show that our approach reduces the number of processed pixels by over 50% without a drop in detection accuracy. The merits of our approach become more significant on a high resolution test set collected from YFCC100M dataset, where our approach maintains high detection performance while reducing the number of processed pixels by about 70% and the detection time by over 50%.