On the Second-order Frechet Derivatives of Eigenvalues of Sturm-Liouville Problems in Potentials

The works of V. A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm-Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and paly an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.Comment: 10 page

Constrained Optimal Querying: Huffman Coding and Beyond

Huffman coding is well known to be useful in certain decision problems involving minimizing the average number of (freely chosen) queries to determine an unknown random variable. However, in problems where the queries are more constrained, the original Huffman coding no longer works. In this paper, we proposed a general model to describe such problems and two code schemes: one is Huffman-based, and the other called GBSC (Greedy Binary Separation Coding). We proved the optimality of GBSC by induction on a binary decision tree, telling us that GBSC is at least as good as Shannon coding. We then compared the two algorithms based on these two codes, by testing them with two problems: DNA detection and 1-player Battleship, and found both to be decent approximating algorithms, with Huffman-based algorithm giving an expected length 1.1 times the true optimal in DNA detection problem, and GBSC yielding an average number of queries 1.4 times the theoretical optimal in 1-player Battleship

Natural Language is All a Graph Needs

The emergence of large-scale pre-trained language models, such as ChatGPT, has revolutionized various research fields in artificial intelligence. Transformers-based large language models (LLMs) have gradually replaced CNNs and RNNs to unify fields of computer vision and natural language processing. Compared with the data that exists relatively independently such as images, videos or texts, graph is a type of data that contains rich structural and relational information. Meanwhile, natural language, as one of the most expressive mediums, excels in describing complex structures. However, existing work on incorporating graph learning problems into the generative language modeling framework remains very limited. As the importance of language models continues to grow, it becomes essential to explore whether LLMs can also replace GNNs as the foundational model for graphs. In this paper, we propose InstructGLM (Instruction-finetuned Graph Language Model), systematically design highly scalable prompts based on natural language instructions, and use natural language to describe the geometric structure and node features of the graph for instruction tuning an LLMs to perform learning and inference on graphs in a generative manner. Our method exceeds all competitive GNN baselines on ogbn-arxiv, Cora and PubMed datasets, which demonstrates the effectiveness of our method and sheds light on generative language models replacing GNNs as the foundation model for graph machine learning.Comment: 21 pages, 2 figures, 5 table

Automaticity in processing spatial-numerical associations: Evidence from a perceptual orientation judgment task of Arabic digits in frames.

Human adults are faster to respond to small/large numerals with their left/right hand when they judge the parity of numerals, which is known as the SNARC (spatial-numerical association of response codes) effect. It has been proposed that the size of the SNARC effect depends on response latencies. The current study introduced a perceptual orientation task, where participants were asked to judge the orientation of a digit or a frame surrounding the digit. The present study first confirmed the SNARC effect with native Chinese speakers (Experiment 1) using a parity task, and then examined whether the emergence and size of the SNARC effect depended on the response latencies (Experiments 2, 3, and 4) using a perceptual orientation judgment task. Our results suggested that (a) the automatic processing of response-related numerical-spatial information occurred with Chinese-speaking participants in the parity task; (b) the SNARC effect was also found when the task did not require semantic access; and (c) the size of the effect depended on the processing speed of the task-relevant dimension. Finally, we proposed an underlying mechanism to explain the SNARC effect in the perceptual orientation judgment task

Deconfounded Causal Collaborative Filtering

Recommender systems may be confounded by various types of confounding factors (also called confounders) that may lead to inaccurate recommendations and sacrificed recommendation performance. Current approaches to solving the problem usually design each specific model for each specific confounder. However, real-world systems may include a huge number of confounders and thus designing each specific model for each specific confounder is unrealistic. More importantly, except for those "explicit confounders" that researchers can manually identify and process such as item's position in the ranking list, there are also many "latent confounders" that are beyond the imagination of researchers. For example, users' rating on a song may depend on their current mood or the current weather, and users' preference on ice creams may depend on the air temperature. Such latent confounders may be unobservable in the recorded training data. To solve the problem, we propose a deconfounded causal collaborative filtering model. We first frame user behaviors with unobserved confounders into a causal graph, and then we design a front-door adjustment model carefully fused with machine learning to deconfound the influence of unobserved confounders. The proposed model is able to handle both global confounders and personalized confounders. Experiments on real-world e-commerce datasets show that our method is able to deconfound unobserved confounders to achieve better recommendation performance.Comment: 9 pages, 5 figures; comments and suggestions are highly appreciate