Extensive experiments have gradually revealed the potential performance bottleneck of modeling Combinatorial Optimization (CO) solving as neural solution prediction tasks. The neural networks, in their pursuit of minimizing the average objective score across the distribution of historical problem instances, diverge from the core target of CO of seeking optimal solutions for every test instance. This calls for an effective search on each problem instance, while the model should serve to provide supporting knowledge that benefits the search. To this end, we propose T2T (Training to Testing) framework that first leverages the generative modeling to estimate the high-quality solution distribution for each instance during training, and then conducts a gradient-based search within the solution space during testing. The proposed neural search paradigm consistently leverages generative modeling, specifically diffusion, for graduated solution improvement. It disrupts the local structure of the given solution by introducing noise and reconstructs a lower-cost solution guided by the optimization objective. Experimental results on Traveling Salesman Problem (TSP) and Maximal Independent Set (MIS) show the significant superiority of T2T, demonstrating an average performance gain of 49.15% for TSP solving and 17.27% for MIS solving compared to the previous state-of-the-art.

In this work we study loss functions for learning and evaluating probability distributions over large discrete domains. Unlike classification or regression where a wide variety of loss functions are used, in the distribution learning and density estimation literature, very few losses outside the dominant \emph{log loss} are applied. We aim to understand this fact, taking an axiomatic approach to the design of loss functions for distributions. We start by proposing a set of desirable criteria that any good loss function should satisfy. Intuitively, these criteria require that the loss function faithfully evaluates a candidate distribution, both in expectation and when estimated on a few samples.

Measurement of spatial fields is of interest in environment monitoring. Recently mobile sensing has been proposed for spatial field reconstruction, which requires a smaller number of sensors when compared to the traditional paradigm of sensing with static sensors. A challenge in mobile sensing is to overcome the location uncertainty of its sensors. While GPS or other localization methods can reduce this uncertainty, we address a more fundamental question: can a location-unaware mobile sensor, recording samples on a directed non-uniform random walk, learn the statistical distribution (as a function of space) of an underlying random process (spatial field)? The answer is in the affirmative for Lipschitz continuous fields, where the accuracy of our distribution-learning method increases with the number of observed field samples (sampling rate). To validate our distribution-learning method, we have created a dataset with 43 experimental trials by measuring sound-level along a fixed path using a location-unaware mobile sound-level meter.

Even humans cannot predict the precise age from a single facial image. They may say that the person is probably in one age group and less likely to be in another.

There are currently two strategies including LDL (label distribution learning) and LE (label enhancement) to predict label distributions.

Markov processes over denumerable products of spaces, describing large systems of automata.