Enabling neural networks to learn complex logical constraints and fulfill symbolic reasoning is a critical challenge. Bridging this gap often requires guiding the neural network’s output distribution to move closer to the symbolic constraints. While diffusion models have shown remarkable generative capability across various domains, we employ the powerful architecture to perform neuro-symbolic learning and solve logical puzzles. Our diffusion-based pipeline adopts a two-stage training strategy: the first stage focuses on cultivating basic reasoning abilities, while the second emphasizes systematic learning of logical constraints. To impose hard constraints on neural outputs in the second stage, we formulate the diffusion reasoner as a Markov decision process and innovatively fine-tune it with an improved proximal policy optimization algorithm. We utilize a rule-based reward signal derived from the logical consistency of neural outputs and adopt a flexible strategy to optimize the diffusion reasoner's policy. We evaluate our methodology on some classical symbolic reasoning benchmarks, including Sudoku, Maze, pathfinding and preference learning. Experimental results demonstrate that our approach achieves outstanding accuracy and logical consistency among neural networks.

The relative similarity testing aims to determine which of the distributions, $P$ or $Q$, is closer to an anchor distribution $U$. Existing kernel-based approaches often test the relative similarity with a fixed kernel in a manually specified alternative hypothesis, e.g., $Q$ is closer to $U$ than $P$. Although kernel selection is known to be important to kernel-based testing methods, the manually specified hypothesis poses a significant challenge for kernel selection in relative similarity testing: Once the hypothesis is specified first, we can always find a kernel such that the hypothesis is rejected. This challenge makes relative similarity testing ill-defined when we want to select a good kernel after the hypothesis is specified. In this paper, we cope with this challenge via learning a proper hypothesis and a kernel simultaneously, instead of learning a kernel after manually specifying the hypothesis. We propose an anchor-based maximum discrepancy (AMD), which defines the relative similarity as the maximum discrepancy between the distances of $(U, P)$ and $(U, Q)$ in a space of deep kernels. Based on AMD, our testing incorporates two phases. In Phase I, we estimate the AMD over the deep kernel space and infer the potential hypothesis. In Phase II, we assess the statistical significance of the potential hypothesis, where we propose a unified testing framework to derive thresholds for tests over different possible hypotheses from Phase I. Lastly, we validate our method theoretically and demonstrate its effectiveness via extensive experiments on benchmark datasets. Codes are publicly available at: https://github.com/tmlr-group/AMD.

To adapt kernel two-sample and independence testing to complex structured data, aggregation of multiple kernels is frequently employed to boost testing power compared to single-kernel tests. However, we observe a phenomenon that directly maximizing multiple kernel-based statistics may result in highly similar kernels that capture highly overlapping information, limiting the effectiveness of aggregation. To address this, we propose an aggregated statistic that explicitly incorporates kernel diversity based on the covariance between different kernels. Moreover, we identify a fundamental challenge: a trade-off between the diversity among kernels and the test power of individual kernels, i.e., the selected kernels should be both effective and diverse. This motivates a testing framework with selection inference, which leverages information from the training phase to select kernels with strong individual performance from the learned diverse kernel pool. We provide rigorous theoretical statements and proofs to show the consistency on the test power and control of Type-I error, along with asymptotic analysis of the proposed statistics. Lastly, we conducted extensive empirical experiments demonstrating the superior performance of our proposed approach across various benchmarks for both two-sample and independence testing.

Learning effective data representations has been crucial in non-parametric two-sample testing. Common approaches will first split data into training and test sets and then learn data representations purely on the training set. However, recent theoretical studies have shown that, as long as the sample indexes are not used during the learning process, the whole data can be used to learn data representations, meanwhile ensuring control of Type-I errors. The above fact motivates us to use the test set (but without sample indexes) to facilitate the data representation learning in the testing. To this end, we propose a representation-learning two-sample testing (RL-TST) framework. RL-TST first performs purely self-supervised representation learning on the entire dataset to capture inherent representations (IRs) that reflect the underlying data manifold. A discriminative model is then trained on these IRs to learn discriminative representations (DRs), enabling the framework to leverage both the rich structural information from IRs and the discriminative power of DRs. Extensive experiments demonstrate that RL-TST outperforms representative approaches by simultaneously using data manifold information in the test set and enhancing test power via finding the DRs with the training set.

Recent years have witnessed increasing attentions on two-sample test with diverse real applications, while this work takes one more step on the exploration of local significant differences for two-sample test. We propose the ME$_\text{MaBiD}$, an effective test for two-sample testing, and the basic idea is to exploit local information by multiple Mahalanobis kernels and introduce bi-directional hypothesis for testing. On the exploration of local significant differences, we first partition the embedding space into several rectangle regions via a new splitting criterion, which is relevant to test power and data correlation. We then explore local significant differences based on our bi-directional masked $p$-value together with the ME$_\text{MaBiD}$ test. Theoretically, we present the asymptotic distribution and lower bounds of test power for our ME$_\text{MaBiD}$ test, and control the familywise error rate on the exploration of local significant differences. We finally conduct extensive experiments to validate the effectiveness of our proposed methods on two-sample test and the exploration of local significant differences.

Measuring the performance of an aeromagnetic compensation system is usually difficult. The standard deviation of the signal has been used as an index in the industry. While the standard deviation is drawn from frequency statistics, it cannot represent the performance on a single sampling point. On the other hand, as the true geomagnetic intensity is unknown, the signal's deviation is actually an approximate measurement of the residual error. This letter first analyzes the traditional neural model for aeromagnetic compensation to reveal the fact that the model can only estimate the expectation of interference. Then, we introduce a stochastic hidden variable to predict the standard deviation synchronously. The proposed model is derived from variational inference and trained as a stochastic gradient variational Bayes estimator. Simulations are performed to show the correlation between the true residual error and the estimated standard deviation.

Conventional acoustic microphones have limited bandwidth, hence inadequate for measuring aero-acoustic waves that require a wide bandwidth from hundreds of kHz to a few MHz and a large pressure dynamic range from ~10Pa to ~1kPa. A lumped-element model is developed and used to design and estimate the static and dynamic mechanical response of an aeroacoustic microphone. Also reported is the implementation and preliminary characterization of such a microphone with a diaphragm supported at the corners by cantilever springs and integrated with piezoresistive transducers made of metal-induced laterally crystallized polycrystalline silicon. A static sensitivity of ~0.1μV/V/Pa and a resonant frequency of 560kHz have been measured.