Oja's algorithm for Streaming Principal Component Analysis (PCA) for $n$ data-points in a $d$ dimensional space achieves the same sin-squared error $O(r_{\mathsf{eff}}/n)$ as the offline algorithm in $O(d)$ space and $O(nd)$ time and a single pass through the datapoints. Here $r_{\mathsf{eff}}$ is the effective rank (ratio of the trace and the principal eigenvalue of the population covariance matrix $\Sigma$). Under this computational budget, we consider the problem of sparse PCA, where the principal eigenvector of $\Sigma$ is $s$-sparse, and $r_{\mathsf{eff}}$ can be large. In this setting, to our knowledge, \textit{there are no known single-pass algorithms} that achieve the minimax error bound in $O(d)$ space and $O(nd)$ time without either requiring strong initialization conditions or assuming further structure (e.g., spiked) of the covariance matrix. We show that a simple single-pass procedure that thresholds the output of Oja's algorithm (the Oja vector) can achieve the minimax error bound under some regularity conditions in $O(d)$ space and $O(nd)$ time. We present a nontrivial and novel analysis of the entries of the unnormalized Oja vector, which involves the projection of a product of independent random matrices on a random initial vector. This is completely different from previous analyses of Oja's algorithm and matrix products, which have been done when the $r_{\mathsf{eff}}$ is bounded.
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A near-field wideband beamforming scheme is investigated for reconfigurable intelligent surface (RIS) assisted multiple-input multiple-output (MIMO) systems, in which a deep learning-based end-to-end (E2E) optimization framework is proposed to maximize the system spectral efficiency. To deal with the near-field double beam split effect, the base station is equipped with frequency-dependent hybrid precoding architecture by introducing sub-connected true time delay (TTD) units, while two specific RIS architectures, namely true time delay-based RIS (TTD-RIS) and virtual subarray-based RIS (SA-RIS), are exploited to realize the frequency-dependent passive beamforming at the RIS. Furthermore, the efficient E2E beamforming models without explicit channel state information are proposed, which jointly exploits the uplink channel training module and the downlink wideband beamforming module. In the proposed network architecture of the E2E models, the classical communication signal processing methods, i.e., polarized filtering and sparsity transform, are leveraged to develop a signal-guided beamforming network. Numerical results show that the proposed E2E models have superior beamforming performance and robustness to conventional beamforming benchmarks. Furthermore, the tradeoff between the beamforming gain and the hardware complexity is investigated for different frequency-dependent RIS architectures, in which the TTD-RIS can achieve better spectral efficiency than the SA-RIS while requiring additional energy consumption and hardware cost.
We investigate the numerical solution of multiscale transport equations using Physics Informed Neural Networks (PINNs) with ReLU activation functions. Therefore, we study the analogy between PINNs and Least-Squares Finite Elements (LSFE) which lies in the shared approach to reformulate the PDE solution as a minimization of a quadratic functional. We prove that in the diffusive regime, the correct limit is not reached, in agreement with known results for first-order LSFE. A diffusive scaling is introduced that can be applied to overcome this, again in full agreement with theoretical results for LSFE. We provide numerical results in the case of slab geometry that support our theoretical findings.
We propose a method for estimating a log-concave density on $\mathbb R^d$ from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave density estimation is hard both computationally and statistically, the independent components assumption alleviates both issues, while still maintaining a large non-parametric class. We prove that under mild conditions, at most $\tilde{\mathcal{O}}(\epsilon^{-4})$ samples (suppressing constants and log factors) suffice for our proposed estimator to be within $\epsilon$ of the original density in squared Hellinger distance. On the computational front, while the usual log-concave maximum likelihood estimate can be obtained via a finite-dimensional convex program, it is slow to compute -- especially in higher dimensions. We demonstrate through numerical experiments that our estimator can be computed efficiently, making it more practical to use.
In a graph bisection problem, we are given a graph $G$ with two equally-sized unlabeled communities, and the goal is to recover the vertices in these communities. A popular heuristic, known as spectral clustering, is to output an estimated community assignment based on the eigenvector corresponding to the second smallest eigenvalue of the Laplacian of $G$. Spectral algorithms can be shown to provably recover the cluster structure for graphs generated from certain probabilistic models, such as the Stochastic Block Model (SBM). However, spectral clustering is known to be non-robust to model mis-specification. Techniques based on semidefinite programming have been shown to be more robust, but they incur significant computational overheads. In this work, we study the robustness of spectral algorithms against semirandom adversaries. Informally, a semirandom adversary is allowed to ``helpfully'' change the specification of the model in a way that is consistent with the ground-truth solution. Our semirandom adversaries in particular are allowed to add edges inside clusters or increase the probability that an edge appears inside a cluster. Semirandom adversaries are a useful tool to determine the extent to which an algorithm has overfit to statistical assumptions on the input. On the positive side, we identify classes of semirandom adversaries under which spectral bisection using the _unnormalized_ Laplacian is strongly consistent, i.e., it exactly recovers the planted partitioning. On the negative side, we show that in these classes spectral bisection with the _normalized_ Laplacian outputs a partitioning that makes a classification mistake on a constant fraction of the vertices. Finally, we demonstrate numerical experiments that complement our theoretical findings.
Schr\"{o}dinger Bridges (SB) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been proposed in the literature, most of these approaches require computationally expensive training schemes, even for solving low-dimensional problems. In this work, we propose an analytic parametrization of a set of feasible policies for steering the distribution of a dynamical system from one Gaussian Mixture Model (GMM) to another. Instead of relying on standard non-convex optimization techniques, the optimal policy within the set can be approximated as the solution of a low-dimensional linear program whose dimension scales linearly with the number of components in each mixture. Furthermore, our method generalizes naturally to more general classes of dynamical systems such as controllable Linear Time-Varying systems that cannot currently be solved using traditional neural SB approaches. We showcase the potential of this approach in low-to-moderate dimensional problems such as image-to-image translation in the latent space of an autoencoder, and various other examples. We also benchmark our approach on an Entropic Optimal Transport (EOT) problem and show that it outperforms state-of-the-art methods in cases where the boundary distributions are mixture models while requiring virtually no training.
Although diffusion models have achieved remarkable success in the field of image generation, their latent space remains under-explored. Current methods for identifying semantics within latent space often rely on external supervision, such as textual information and segmentation masks. In this paper, we propose a method to identify semantic attributes in the latent space of pre-trained diffusion models without any further training. By projecting the Jacobian of the targeted semantic region into a low-dimensional subspace which is orthogonal to the non-masked regions, our approach facilitates precise semantic discovery and control over local masked areas, eliminating the need for annotations. We conducted extensive experiments across multiple datasets and various architectures of diffusion models, achieving state-of-the-art performance. In particular, for some specific face attributes, the performance of our proposed method even surpasses that of supervised approaches, demonstrating its superior ability in editing local image properties.
Contextualized embeddings vary by context, even for the same token, and form a distribution in the embedding space. To analyze this distribution, we focus on the norm of the mean embedding and the variance of the embeddings. In this study, we first demonstrate that these values follow the well-known formula for variance in statistics and provide an efficient sequential computation method. Then, by observing embeddings from intermediate layers of several Transformer models, we found a strong trade-off relationship between the norm and the variance: as the mean embedding becomes closer to the origin, the variance increases. This trade-off is likely influenced by the layer normalization mechanism used in Transformer models. Furthermore, when the sets of token embeddings are treated as clusters, we show that the variance of the entire embedding set can theoretically be decomposed into the within-cluster variance and the between-cluster variance. We found experimentally that as the layers of Transformer models deepen, the embeddings move farther from the origin, the between-cluster variance relatively decreases, and the within-cluster variance relatively increases. These results are consistent with existing studies on the anisotropy of the embedding spaces across layers.
Retrieval-Augmented Generation (RAG) merges retrieval methods with deep learning advancements to address the static limitations of large language models (LLMs) by enabling the dynamic integration of up-to-date external information. This methodology, focusing primarily on the text domain, provides a cost-effective solution to the generation of plausible but incorrect responses by LLMs, thereby enhancing the accuracy and reliability of their outputs through the use of real-world data. As RAG grows in complexity and incorporates multiple concepts that can influence its performance, this paper organizes the RAG paradigm into four categories: pre-retrieval, retrieval, post-retrieval, and generation, offering a detailed perspective from the retrieval viewpoint. It outlines RAG's evolution and discusses the field's progression through the analysis of significant studies. Additionally, the paper introduces evaluation methods for RAG, addressing the challenges faced and proposing future research directions. By offering an organized framework and categorization, the study aims to consolidate existing research on RAG, clarify its technological underpinnings, and highlight its potential to broaden the adaptability and applications of LLMs.
In multi-turn dialog, utterances do not always take the full form of sentences \cite{Carbonell1983DiscoursePA}, which naturally makes understanding the dialog context more difficult. However, it is essential to fully grasp the dialog context to generate a reasonable response. Hence, in this paper, we propose to improve the response generation performance by examining the model's ability to answer a reading comprehension question, where the question is focused on the omitted information in the dialog. Enlightened by the multi-task learning scheme, we propose a joint framework that unifies these two tasks, sharing the same encoder to extract the common and task-invariant features with different decoders to learn task-specific features. To better fusing information from the question and the dialog history in the encoding part, we propose to augment the Transformer architecture with a memory updater, which is designed to selectively store and update the history dialog information so as to support downstream tasks. For the experiment, we employ human annotators to write and examine a large-scale dialog reading comprehension dataset. Extensive experiments are conducted on this dataset, and the results show that the proposed model brings substantial improvements over several strong baselines on both tasks. In this way, we demonstrate that reasoning can indeed help better response generation and vice versa. We release our large-scale dataset for further research.
Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.
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