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Principal Component Analysis (PCA) is one of the most commonly used statistical methods for data exploration, and for dimensionality reduction wherein the first few principal components account for an appreciable proportion of the variability in the data. Less commonly, attention is paid to the last principal components because they do not account for an appreciable proportion of variability. However, this defining characteristic of the last principal components also qualifies them as combinations of variables that are constant across the cases. Such constant-combinations are important because they may reflect underlying laws of nature. In situations involving a large number of noisy covariates, the underlying law may not correspond to the last principal component, but rather to one of the last. Consequently, a criterion is required to identify the relevant eigenvector. In this paper, two examples are employed to demonstrate the proposed methodology; one from Physics, involving a small number of covariates, and another from Meteorology wherein the number of covariates is in the thousands. It is shown that with an appropriate selection criterion, PCA can be employed to ``discover" Kepler's third law (in the former), and the hypsometric equation (in the latter).

相關內容

在統計中(zhong),主成分分析(xi)(PCA)是一(yi)種(zhong)通(tong)過最大化每個(ge)維度的(de)(de)方(fang)差來將較高維度空間中(zhong)的(de)(de)數(shu)據投(tou)影到(dao)較低維度空間中(zhong)的(de)(de)方(fang)法。給定二(er)維,三維或更高維空間中(zhong)的(de)(de)點集合(he),可(ke)以(yi)將“最佳(jia)擬合(he)”線(xian)(xian)定義為最小化從點到(dao)線(xian)(xian)的(de)(de)平(ping)均平(ping)方(fang)距離的(de)(de)線(xian)(xian)。可(ke)以(yi)從垂(chui)直于(yu)第一(yi)條直線(xian)(xian)的(de)(de)方(fang)向類似(si)地選擇下一(yi)條最佳(jia)擬合(he)線(xian)(xian)。重(zhong)復此過程會產生一(yi)個(ge)正交的(de)(de)基礎,其(qi)中(zhong)數(shu)據的(de)(de)不同單個(ge)維度是不相關(guan)的(de)(de)。 這些(xie)基向量稱為主成分。

Mitigating biases in machine learning models has gained increasing attention in Natural Language Processing (NLP). Yet, only a few studies focus on fair text embeddings, which are crucial yet challenging for real-world applications. In this paper, we propose a novel method for learning fair text embeddings. We achieve fairness while maintaining utility trade-off by ensuring conditional independence between sensitive attributes and text embeddings conditioned on the content. Specifically, we enforce that embeddings of texts with different sensitive attributes but identical content maintain the same distance toward the embedding of their corresponding neutral text. Furthermore, we address the issue of lacking proper training data by using Large Language Models (LLMs) to augment texts into different sensitive groups. Our extensive evaluations demonstrate that our approach effectively improves fairness while preserving the utility of embeddings, representing a pioneering effort in achieving conditional independence for fair text embeddings.

Vision Transformers (ViTs) have achieved remarkable success in computer vision tasks. However, their potential in rotation-sensitive scenarios has not been fully explored, and this limitation may be inherently attributed to the lack of spatial invariance in the data-forwarding process. In this study, we present a novel approach, termed Spatial Transform Decoupling (STD), providing a simple-yet-effective solution for oriented object detection with ViTs. Built upon stacked ViT blocks, STD utilizes separate network branches to predict the position, size, and angle of bounding boxes, effectively harnessing the spatial transform potential of ViTs in a divide-and-conquer fashion. Moreover, by aggregating cascaded activation masks (CAMs) computed upon the regressed parameters, STD gradually enhances features within regions of interest (RoIs), which complements the self-attention mechanism. Without bells and whistles, STD achieves state-of-the-art performance on the benchmark datasets including DOTA-v1.0 (82.24% mAP) and HRSC2016 (98.55% mAP), which demonstrates the effectiveness of the proposed method. Source code is available at //github.com/yuhongtian17/Spatial-Transform-Decoupling.

Matrix factorization (MF) is a simple collaborative filtering technique that achieves superior recommendation accuracy by decomposing the user-item interaction matrix into user and item latent matrices. Because the model typically learns each interaction independently, it may overlook the underlying shared dependencies between users and items, resulting in less stable and interpretable recommendations. Based on these insights, we propose "Hierarchical Matrix Factorization" (HMF), which incorporates clustering concepts to capture the hierarchy, where leaf nodes and other nodes correspond to users/items and clusters, respectively. Central to our approach, called hierarchical embeddings, is the additional decomposition of the latent matrices (embeddings) into probabilistic connection matrices, which link the hierarchy, and a root cluster latent matrix. The embeddings are differentiable, allowing simultaneous learning of interactions and clustering using a single gradient descent method. Furthermore, the obtained cluster-specific interactions naturally summarize user-item interactions and provide interpretability. Experimental results on ratings and ranking predictions show that HMF outperforms existing MF methods, in particular achieving a 1.37 point improvement in RMSE for sparse interactions. Additionally, it was confirmed that the clustering integration of HMF has the potential for faster learning convergence and mitigation of overfitting compared to MF, and also provides interpretability through a cluster-centered case study.

Graph Neural Networks (GNNs) have been successfully used in many problems involving graph-structured data, achieving state-of-the-art performance. GNNs typically employ a message-passing scheme, in which every node aggregates information from its neighbors using a permutation-invariant aggregation function. Standard well-examined choices such as the mean or sum aggregation functions have limited capabilities, as they are not able to capture interactions among neighbors. In this work, we formalize these interactions using an information-theoretic framework that notably includes synergistic information. Driven by this definition, we introduce the Graph Ordering Attention (GOAT) layer, a novel GNN component that captures interactions between nodes in a neighborhood. This is achieved by learning local node orderings via an attention mechanism and processing the ordered representations using a recurrent neural network aggregator. This design allows us to make use of a permutation-sensitive aggregator while maintaining the permutation-equivariance of the proposed GOAT layer. The GOAT model demonstrates its increased performance in modeling graph metrics that capture complex information, such as the betweenness centrality and the effective size of a node. In practical use-cases, its superior modeling capability is confirmed through its success in several real-world node classification benchmarks.

With the rapid development of deep learning, training Big Models (BMs) for multiple downstream tasks becomes a popular paradigm. Researchers have achieved various outcomes in the construction of BMs and the BM application in many fields. At present, there is a lack of research work that sorts out the overall progress of BMs and guides the follow-up research. In this paper, we cover not only the BM technologies themselves but also the prerequisites for BM training and applications with BMs, dividing the BM review into four parts: Resource, Models, Key Technologies and Application. We introduce 16 specific BM-related topics in those four parts, they are Data, Knowledge, Computing System, Parallel Training System, Language Model, Vision Model, Multi-modal Model, Theory&Interpretability, Commonsense Reasoning, Reliability&Security, Governance, Evaluation, Machine Translation, Text Generation, Dialogue and Protein Research. In each topic, we summarize clearly the current studies and propose some future research directions. At the end of this paper, we conclude the further development of BMs in a more general view.

Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.

Graph Neural Networks (GNN) is an emerging field for learning on non-Euclidean data. Recently, there has been increased interest in designing GNN that scales to large graphs. Most existing methods use "graph sampling" or "layer-wise sampling" techniques to reduce training time. However, these methods still suffer from degrading performance and scalability problems when applying to graphs with billions of edges. This paper presents GBP, a scalable GNN that utilizes a localized bidirectional propagation process from both the feature vectors and the training/testing nodes. Theoretical analysis shows that GBP is the first method that achieves sub-linear time complexity for both the precomputation and the training phases. An extensive empirical study demonstrates that GBP achieves state-of-the-art performance with significantly less training/testing time. Most notably, GBP can deliver superior performance on a graph with over 60 million nodes and 1.8 billion edges in less than half an hour on a single machine.

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.

Embedding entities and relations into a continuous multi-dimensional vector space have become the dominant method for knowledge graph embedding in representation learning. However, most existing models ignore to represent hierarchical knowledge, such as the similarities and dissimilarities of entities in one domain. We proposed to learn a Domain Representations over existing knowledge graph embedding models, such that entities that have similar attributes are organized into the same domain. Such hierarchical knowledge of domains can give further evidence in link prediction. Experimental results show that domain embeddings give a significant improvement over the most recent state-of-art baseline knowledge graph embedding models.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.

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