We consider the problem of learning an arbitrarily-biased ReLU activation (or neuron) over Gaussian marginals with the squared loss objective. Despite the ReLU neuron being the basic building block of modern neural networks, we still do not understand the basic algorithmic question of whether one arbitrary ReLU neuron is learnable in the non-realizable setting. In particular, all existing polynomial time algorithms only provide approximation guarantees for the better-behaved unbiased setting or restricted bias setting. Our main result is a polynomial time statistical query (SQ) algorithm that gives the first constant factor approximation for arbitrary bias. It outputs a ReLU activation that achieves a loss of $O(\mathrm{OPT}) + \varepsilon$ in time $\mathrm{poly}(d,1/\varepsilon)$, where $\mathrm{OPT}$ is the loss obtained by the optimal ReLU activation. Our algorithm presents an interesting departure from existing algorithms, which are all based on gradient descent and thus fall within the class of correlational statistical query (CSQ) algorithms. We complement our algorithmic result by showing that no polynomial time CSQ algorithm can achieve a constant factor approximation. Together, these results shed light on the intrinsic limitation of gradient descent, while identifying arguably the simplest setting (a single neuron) where there is a separation between SQ and CSQ algorithms.

Spatial networks are useful for modeling geographic phenomena where spatial interaction plays an important role. To analyze the spatial networks and their internal structures, graph-based methods such as community detection have been widely used. Community detection aims to extract strongly connected components from the network and reveal the hidden relationships between nodes, but they usually do not involve the attribute information. To consider edge-based interactions and node attributes together, this study proposed a family of GeoAI-enhanced unsupervised community detection methods called region2vec based on Graph Attention Networks (GAT) and Graph Convolutional Networks (GCN). The region2vec methods generate node neural embeddings based on attribute similarity, geographic adjacency and spatial interactions, and then extract network communities based on node embeddings using agglomerative clustering. The proposed GeoAI-based methods are compared with multiple baselines and perform the best when one wants to maximize node attribute similarity and spatial interaction intensity simultaneously within the spatial network communities. It is further applied in the shortage area delineation problem in public health and demonstrates its promise in regionalization problems.

As the availability of financial services online continues to grow, the incidence of fraud has surged correspondingly. Fraudsters continually seek new and innovative ways to circumvent the detection algorithms in place. Traditionally, fraud detection relied on rule-based methods, where rules were manually created based on transaction data features. However, these techniques soon became ineffective due to their reliance on manual rule creation and their inability to detect complex data patterns. Today, a significant portion of the financial services sector employs various machine learning algorithms, such as XGBoost, Random Forest, and neural networks, to model transaction data. While these techniques have proven more efficient than rule-based methods, they still fail to capture interactions between different transactions and their interrelationships. Recently, graph-based techniques have been adopted for financial fraud detection, leveraging graph topology to aggregate neighborhood information of transaction data using Graph Neural Networks (GNNs). Despite showing improvements over previous methods, these techniques still struggle to keep pace with the evolving camouflaging tactics of fraudsters and suffer from information loss due to over-smoothing. In this paper, we propose a novel algorithm that employs an efficient neighborhood sampling method, effective for camouflage detection and preserving crucial feature information from non-similar nodes. Additionally, we introduce a novel GNN architecture that utilizes attention mechanisms and preserves holistic neighborhood information to prevent information loss. We test our algorithm on financial data to show that our method outperforms other state-of-the-art graph algorithms.

We study the problem of learning feature representations from a pair of random variables, where we focus on the representations that are induced by their dependence. We provide sufficient and necessary conditions for such dependence induced representations, and illustrate their connections to Hirschfeld--Gebelein--R\'{e}nyi (HGR) maximal correlation functions and minimal sufficient statistics. We characterize a large family of loss functions that can learn dependence induced representations, including cross entropy, hinge loss, and their regularized variants. In particular, we show that the features learned from this family can be expressed as the composition of a loss-dependent function and the maximal correlation function, which reveals a key connection between representations learned from different losses. Our development also gives a statistical interpretation of the neural collapse phenomenon observed in deep classifiers. Finally, we present the learning design based on the feature separation, which allows hyperparameter tuning during inference.

In recent years, the field of machine learning has witnessed significant advancements, particularly in the development of optimisation algorithms that enhance the efficiency and effectiveness of training deep neural networks. Among these algorithms, the Adam optimiser has gained widespread popularity due to its adaptive learning rate capabilities, which enable more efficient convergence compared to traditional methods such as stochastic gradient descent. However, despite its advantages, Adam is not without its limitations, particularly when it comes to handling issues such as overshooting and stagnation during the training process. To address these challenges, we introduce AdamZ as an advanced variant of the Adam optimiser. AdamZ is specifically designed to dynamically adjust the learning rate responsive to the characteristics of the loss function, thereby improving both convergence stability and model accuracy. This novel optimiser integrates mechanisms to detect and mitigate overshooting, at the point where the optimiser has stepped too far into the parameter space, and stagnation at points, where progress has started to stall despite ongoing training. By introducing hyperparameters such as overshoot and stagnation factors, thresholds, and patience levels, AdamZ provides a more responsive approach to learning rate adaptation than obtained through Adam.

LoRA has emerged as one of the de facto methods for fine-tuning foundation models with low computational cost and memory footprint. The idea is to only train a low-rank perturbation to the weights of a pre-trained model, given supervised data for a downstream task. Despite its empirical sucess, from a mathematical perspective it remains poorly understood what learning mechanisms ensure that gradient descent converges to useful low-rank perturbations. In this work we study low-rank fine-tuning in a student-teacher setting. We are given the weights of a two-layer base model $f$, as well as i.i.d. samples $(x,f^*(x))$ where $x$ is Gaussian and $f^*$ is the teacher model given by perturbing the weights of $f$ by a rank-1 matrix. This generalizes the setting of generalized linear model (GLM) regression where the weights of $f$ are zero. When the rank-1 perturbation is comparable in norm to the weight matrix of $f$, the training dynamics are nonlinear. Nevertheless, in this regime we prove under mild assumptions that a student model which is initialized at the base model and trained with online gradient descent will converge to the teacher in $dk^{O(1)}$ iterations, where $k$ is the number of neurons in $f$. Importantly, unlike in the GLM setting, the complexity does not depend on fine-grained properties of the activation's Hermite expansion. We also prove that in our setting, learning the teacher model "from scratch'' can require significantly more iterations.

Semi-supervised learning (SSL) has made notable advancements in medical image segmentation (MIS), particularly in scenarios with limited labeled data and significantly enhancing data utilization efficiency. Previous methods primarily focus on complex training strategies to utilize unlabeled data but neglect the importance of graph structural information. Different from existing methods, we propose a graph-based clustering for semi-supervised medical image segmentation (GraphCL) by jointly modeling graph data structure in a unified deep model. The proposed GraphCL model enjoys several advantages. Firstly, to the best of our knowledge, this is the first work to model the data structure information for semi-supervised medical image segmentation (SSMIS). Secondly, to get the clustered features across different graphs, we integrate both pairwise affinities between local image features and raw features as inputs. Extensive experimental results on three standard benchmarks show that the proposed GraphCL algorithm outperforms state-of-the-art semi-supervised medical image segmentation methods.

This paper presents the first Open-Vocabulary Online 3D semantic SLAM pipeline, that we denote as OVO-SLAM. Our primary contribution is in the pipeline itself, particularly in the mapping thread. Given a set of posed RGB-D frames, we detect and track 3D segments, which we describe using CLIP vectors, calculated through a novel aggregation from the viewpoints where these 3D segments are observed. Notably, our OVO-SLAM pipeline is not only faster but also achieves better segmentation metrics compared to offline approaches in the literature. Along with superior segmentation performance, we show experimental results of our contributions integrated with Gaussian-SLAM, being the first ones demonstrating end-to-end open-vocabulary online 3D reconstructions without relying on ground-truth camera poses or scene geometry.

Equivariant Imaging (EI) regularization has become the de-facto technique for unsupervised training of deep imaging networks, without any need of ground-truth data. Observing that the EI-based unsupervised training paradigm currently has significant computational redundancy leading to inefficiency in high-dimensional applications, we propose a sketched EI regularization which leverages the randomized sketching techniques for acceleration. We then extend our sketched EI regularization to develop an accelerated deep internal learning framework -- Sketched Equivariant Deep Image Prior (Sk-EI-DIP), which can be efficiently applied for single-image and task-adapted reconstruction. Additionally, for network adaptation tasks, we propose a parameter-efficient approach for accelerating both EI-DIP and Sk-EI-DIP via optimizing only the normalization layers. Our numerical study on X-ray CT image reconstruction tasks demonstrate that our approach can achieve order-of-magnitude computational acceleration over standard EI-based counterpart in single-input setting, and network adaptation at test time.

Transformers have demonstrated great power in the recent development of large foundational models. In particular, the Vision Transformer (ViT) has brought revolutionary changes to the field of vision, achieving significant accomplishments on the experimental side. However, their theoretical capabilities, particularly in terms of generalization when trained to overfit training data, are still not fully understood. To address this gap, this work delves deeply into the benign overfitting perspective of transformers in vision. To this end, we study the optimization of a Transformer composed of a self-attention layer with softmax followed by a fully connected layer under gradient descent on a certain data distribution model. By developing techniques that address the challenges posed by softmax and the interdependent nature of multiple weights in transformer optimization, we successfully characterized the training dynamics and achieved generalization in post-training. Our results establish a sharp condition that can distinguish between the small test error phase and the large test error regime, based on the signal-to-noise ratio in the data model. The theoretical results are further verified by experimental simulation. To the best of our knowledge, this is the first work to characterize benign overfitting for Transformers.