Recently, there has been a growing interest in developing saliency methods that provide visual explanations of network predictions. Still, the usability of existing methods is limited to image classification models. To overcome this limitation, we extend the existing approaches to generate grid saliencies, which provide spatially coherent visual explanations for (pixel-level) dense prediction networks. As the proposed grid saliency allows to spatially disentangle the object and its context, we specifically explore its potential to produce context explanations for semantic segmentation networks, discovering which context most influences the class predictions inside a target object area. We investigate the effectiveness of grid saliency on a synthetic dataset with an artificially induced bias between objects and their context as well as on the real-world Cityscapes dataset using state-of-the-art segmentation networks. Our results show that grid saliency can be successfully used to provide easily interpretable context explanations and, moreover, can be employed for detecting and localizing contextual biases present in the data.

Large Language Models (LLMs) have transformed natural language processing and extended their powerful capabilities to multi-modal domains. As LLMs continue to advance, it is crucial to develop diverse and appropriate metrics for their evaluation. In this paper, we introduce a novel rank-based metric, Diff-eRank, grounded in information theory and geometry principles. Diff-eRank assesses LLMs by analyzing their hidden representations, providing a quantitative measure of how efficiently they eliminate redundant information during training. We demonstrate the applicability of Diff-eRank in both single-modal (e.g., language) and multimodal settings. For language models, our results show that Diff-eRank increases with model size and correlates well with conventional metrics such as loss and accuracy. In the multi-modal context, we propose an alignment evaluation method based on the eRank, and verify that contemporary multi-modal LLMs exhibit strong alignment performance based on our method.

Deep learning has seen remarkable developments over the last years, many of them inspired by neuroscience. However, the main learning mechanism behind these advances - error backpropagation - appears to be at odds with neurobiology. Here, we introduce a multilayer neuronal network model with simplified dendritic compartments in which error-driven synaptic plasticity adapts the network towards a global desired output. In contrast to previous work our model does not require separate phases and synaptic learning is driven by local dendritic prediction errors continuously in time. Such errors originate at apical dendrites and occur due to a mismatch between predictive input from lateral interneurons and activity from actual top-down feedback. Through the use of simple dendritic compartments and different cell-types our model can represent both error and normal activity within a pyramidal neuron. We demonstrate the learning capabilities of the model in regression and classification tasks, and show analytically that it approximates the error backpropagation algorithm. Moreover, our framework is consistent with recent observations of learning between brain areas and the architecture of cortical microcircuits. Overall, we introduce a novel view of learning on dendritic cortical circuits and on how the brain may solve the long-standing synaptic credit assignment problem.

Quantum computing pioneer D-Wave Quantum on Tuesday announced the general availability of its sixth-generation quantum computer, the Advantage2. The company said the Advantage2 offers orders-of-magnitude greater performance compared to its prior system, expanding the tasks the company can accomplish in optimization problems. The machine even achieves the long-sought goal of quantum "supremacy," says the company, despite that term's highly controversial past. "This is a really historic moment for both D-Wave and the quantum computing industry," said D-Wave CEO Alan Baratz in an interview via Zoom. "Fundamentally, our technology is doing something that can't be touched classically."

Graph neural networks (GNNs) have recently emerged as powerful tools for addressing complex optimization problems. It has been theoretically demonstrated that GNNs can universally approximate the solution mapping functions of linear programming (LP) problems. However, these theoretical results typically require GNNs to have large parameter sizes. Conversely, empirical experiments have shown that relatively small GNNs can solve LPs effectively, revealing a significant discrepancy between theoretical predictions and practical observations. In this work, we aim to bridge this gap by providing a theoretical foundation for the effectiveness of smaller GNNs. We prove that polylogarithmic-depth, constant-width GNNs are sufficient to solve packing and covering LPs, two widely used classes of LPs. Our proof leverages the capability of GNNs to simulate a variant of the gradient descent algorithm on a carefully selected potential function. Additionally, we introduce a new GNN architecture, termed GD-Net. Experimental results demonstrate that GD-Net significantly outperforms conventional GNN structures while using fewer parameters.

Transformer has shown state-of-the-art performance on various applications and has recently emerged as a promising tool for surrogate modeling of partial differential equations (PDEs). Despite the introduction of linear-complexity attention, applying Transformer to problems with a large number of grid points can be numerically unstable and computationally expensive. In this work, we propose Factorized Transformer (FactFormer), which is based on an axial factorized kernel integral. Concretely, we introduce a learnable projection operator that decomposes the input function into multiple sub-functions with one-dimensional domain. These sub-functions are then evaluated and used to compute the instance-based kernel with an axial factorized scheme. We showcase that the proposed model is able to simulate 2D Kolmogorov flow on a 256 256 grid and 3D smoke buoyancy on a 64 64 64 grid with good accuracy and efficiency. The proposed factorized scheme can serve as a computationally efficient low-rank surrogate for the full attention scheme when dealing with multi-dimensional problems.

Topological features based on persistent homology can capture high-order structural information which can then be used to augment graph neural network methods. However, computing extended persistent homology summaries remains slow for large and dense graphs and can be a serious bottleneck for the learning pipeline. Inspired by recent success in neural algorithmic reasoning, we propose a novel graph neural network to estimate extended persistence diagrams (EPDs) on graphs efficiently. Our model is built on algorithmic insights, and benefits from better supervision and closer alignment with the EPD computation algorithm.

In this paper, we study the statistical limits in terms of Sobolev norms of gradient descent for solving inverse problems from randomly sampled noisy observations using a general class of objective functions. Our class of objective functions includes Sobolev training for kernel regression, Deep Ritz Methods (DRM), and Physics Informed Neural Networks (PINN) for solving elliptic partial differential equations (PDEs) as special cases. We consider a potentially infinite-dimensional parameterization of our model using a suitable Reproducing Kernel Hilbert Space and a continuous parameterization of problem hardness through the definition of kernel integral operators. We prove that gradient descent over this objective function can also achieve statistical optimality and the optimal number of passes over the data increases with sample size. Based on our theory, we explain an implicit acceleration of using a Sobolev norm as the objective function for training, inferring that the optimal number of epochs of DRM becomes larger than the number of PINN when both the data size and the hardness of tasks increase, although both DRM and PINN can achieve statistical optimality.

We address the problem of optimizing over functions defined on node subsets in a graph. The optimization of such functions is often a non-trivial task given their combinatorial, black-box and expensive-to-evaluate nature. Although various algorithms have been introduced in the literature, most are either task-specific or computationally inefficient and only utilize information about the graph structure without considering the characteristics of the function. To address these limitations, we utilize Bayesian Optimization (BO), a sample-efficient black-box solver, and propose a novel framework for combinatorial optimization on graphs. More specifically, we map each k-node subset in the original graph to a node in a new combinatorial graph and adopt a local modeling approach to efficiently traverse the latter graph by progressively sampling its subgraphs using a recursive algorithm. Extensive experiments under both synthetic and real-world setups demonstrate the effectiveness of the proposed BO framework on various types of graphs and optimization tasks, where its behavior is analyzed in detail with ablation studies.

Large scale neural networks used in practice are highly over-parameterized with far more trainable model parameters compared to the number of training examples. Consequently, optimization objectives for learning such high capacity models have many global minima that fit training data perfectly. However, minimizing the training loss using specific optimization algorithms take us to not just any global minima, but some special global minima, e.g., global minima minimizing some regularizer R(β). In over-parameterized models, specially deep neural networks, much, if not most, of the inductive bias of the learned model comes from this implicit regularization from the optimization algorithm. Understanding the implicit bias, e.g., via characterizing R(β), is thus essential for understanding how and what the model learns.