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Graph Neural Networks (GNNs) are the state-of-the-art model for machine learning on graph-structured data. The most popular class of GNNs operate by exchanging information between adjacent nodes, and are known as Message Passing Neural Networks (MPNNs). Given their widespread use, understanding the expressive power of MPNNs is a key question. However, existing results typically consider settings with uninformative node features. In this paper, we provide a rigorous analysis to determine which function classes of node features can be learned by an MPNN of a given capacity. We do so by measuring the level of pairwise interactions between nodes that MPNNs allow for. This measure provides a novel quantitative characterization of the so-called over-squashing effect, which is observed to occur when a large volume of messages is aggregated into fixed-size vectors. Using our measure, we prove that, to guarantee sufficient communication between pairs of nodes, the capacity of the MPNN must be large enough, depending on properties of the input graph structure, such as commute times. For many relevant scenarios, our analysis results in impossibility statements in practice, showing that over-squashing hinders the expressive power of MPNNs. We validate our theoretical findings through extensive controlled experiments and ablation studies.

Image blending aims to combine multiple images seamlessly. It remains challenging for existing 2D-based methods, especially when input images are misaligned due to differences in 3D camera poses and object shapes. To tackle these issues, we propose a 3D-aware blending method using generative Neural Radiance Fields (NeRF), including two key components: 3D-aware alignment and 3D-aware blending. For 3D-aware alignment, we first estimate the camera pose of the reference image with respect to generative NeRFs and then perform 3D local alignment for each part. To further leverage 3D information of the generative NeRF, we propose 3D-aware blending that directly blends images on the NeRF's latent representation space, rather than raw pixel space. Collectively, our method outperforms existing 2D baselines, as validated by extensive quantitative and qualitative evaluations with FFHQ and AFHQ-Cat.

Medical imaging models have been shown to encode information about patient demographics such as age, race, and sex in their latent representation, raising concerns about their potential for discrimination. Here, we ask whether requiring models not to encode demographic attributes is desirable. We point out that marginal and class-conditional representation invariance imply the standard group fairness notions of demographic parity and equalized odds, respectively, while additionally requiring risk distribution matching, thus potentially equalizing away important group differences. Enforcing the traditional fairness notions directly instead does not entail these strong constraints. Moreover, representationally invariant models may still take demographic attributes into account for deriving predictions. The latter can be prevented using counterfactual notions of (individual) fairness or invariance. We caution, however, that properly defining medical image counterfactuals with respect to demographic attributes is highly challenging. Finally, we posit that encoding demographic attributes may even be advantageous if it enables learning a task-specific encoding of demographic features that does not rely on social constructs such as 'race' and 'gender.' We conclude that demographically invariant representations are neither necessary nor sufficient for fairness in medical imaging. Models may need to encode demographic attributes, lending further urgency to calls for comprehensive model fairness assessments in terms of predictive performance across diverse patient groups.

We discuss the problem of estimating Radon-Nikodym derivatives. This problem appears in various applications, such as covariate shift adaptation, likelihood-ratio testing, mutual information estimation, and conditional probability estimation. To address the above problem, we employ the general regularization scheme in reproducing kernel Hilbert spaces. The convergence rate of the corresponding regularized algorithm is established by taking into account both the smoothness of the derivative and the capacity of the space in which it is estimated. This is done in terms of general source conditions and the regularized Christoffel functions. We also find that the reconstruction of Radon-Nikodym derivatives at any particular point can be done with high order of accuracy. Our theoretical results are illustrated by numerical simulations.

In Bayesian inference, a simple and popular approach to reduce the burden of computing high dimensional integrals against a posterior $\pi$ is to make the Laplace approximation $\hat\gamma$. This is a Gaussian distribution, so computing $\int fd\pi$ via the approximation $\int fd\hat\gamma$ is significantly less expensive. In this paper, we make two general contributions to the topic of high-dimensional Laplace approximations, as well as a third contribution specific to a logistic regression model. First, we tighten the dimension dependence of the error $|\int fd\pi - \int fd\hat\gamma|$ for a broad class of functions $f$. Second, we derive a higher-accuracy approximation $\hat\gamma_S$ to $\pi$, which is a skew-adjusted modification to $\hat\gamma$. Our third contribution - in the setting of Bayesian inference for logistic regression with Gaussian design - is to use the first two results to derive upper bounds which hold uniformly over different sample realizations, and lower bounds on the Laplace mean approximation error. In particular, we prove a skewed Bernstein-von Mises Theorem in this logistic regression setting.

We investigate non-wellfounded proof systems based on parsimonious logic, a weaker variant of linear logic where the exponential modality ! is interpreted as a constructor for streams over finite data. Logical consistency is maintained at a global level by adapting a standard progressing criterion. We present an infinitary version of cut-elimination based on finite approximations, and we prove that, in presence of the progressing criterion, it returns well-defined non-wellfounded proofs at its limit. Furthermore, we show that cut-elimination preserves the progressive criterion and various regularity conditions internalizing degrees of proof-theoretical uniformity. Finally, we provide a denotational semantics for our systems based on the relational model.

Benchmarking and evaluating deep learning models and systems necessitate a meticulous approach to ensure comprehensive assessment. In practical applications, it is paramount to consider both the inference quality and the inference time, particularly within critical contexts, where stringent requirements demand the simultaneous satisfaction of both metrics. Neglecting either aspect can result in severe and irreversible consequences, including loss of human life and property damage. Unfortunately, many studies lack a comprehensive consideration of these metrics, often conducted under ideal or permissive conditions, thereby leading to incomplete or non-intuitive evaluation methodologies. This study reveals that deep learning inference quality exhibits fluctuations, which further introduces complications and challenges to the benchmarking and evaluation. To better characterize the phenomenon, the concept of "tail quality" is introduced, which indicates the quality at the tail of distributions. "Tail quality" can offer a more objective evaluation, overcoming the limitations of conventional inference quality and inference time metrics in capturing the quality fluctuation phenomenon. To capture the phenomenon, this paper also proposes a pioneering evaluation framework for comprehensive assessment and analysis of various factors affecting inference time and quality. Leveraging this framework enables the anticipation of the potential distribution of inference time and inference quality, thus capturing "tail quality" before practically applying deep learning. The effectiveness of the evaluation framework is validated through experiments conducted on deep learning models for three different tasks across four systems. Furthermore, employing this evaluation framework, the experiments conducted a preliminary analysis of several factors influencing inference quality and inference time.

The research in this article aims to find conditions of an algorithmic nature that are necessary and sufficient to transform any Boolean function in conjunctive normal form into a specific form that guarantees the satisfiability of this function. To find such conditions, we use the concept of a special covering of a set introduced in [13], and investigate the connection between this concept and the notion of satisfiability of Boolean functions. As shown, the problem of existence of a special covering for a set is equivalent to the Boolean satisfiability problem. Thus, an important result is the proof of the existence of necessary and sufficient conditions that make it possible to find out if there is a special covering for the set under the special decomposition. This result allows us to formulate the necessary and sufficient algorithmic conditions for Boolean satisfiability, considering the function in conjunctive normal form as a set of clauses. In parallel, as a result of the aforementioned algorithmic procedure, we obtain the values of the variables that ensure the satisfiability of this function. The terminology used related to graph theory, set theory, Boolean functions and complexity theory is consistent with the terminology in [1], [2], [3], [4]. The newly introduced terms are not found in use by other authors and do not contradict to other terms.

The LSTM network was proposed to overcome the difficulty in learning long-term dependence, and has made significant advancements in applications. With its success and drawbacks in mind, this paper raises the question - do RNN and LSTM have long memory? We answer it partially by proving that RNN and LSTM do not have long memory from a statistical perspective. A new definition for long memory networks is further introduced, and it requires the model weights to decay at a polynomial rate. To verify our theory, we convert RNN and LSTM into long memory networks by making a minimal modification, and their superiority is illustrated in modeling long-term dependence of various datasets.

We study the problem of named entity recognition (NER) from electronic medical records, which is one of the most fundamental and critical problems for medical text mining. Medical records which are written by clinicians from different specialties usually contain quite different terminologies and writing styles. The difference of specialties and the cost of human annotation makes it particularly difficult to train a universal medical NER system. In this paper, we propose a label-aware double transfer learning framework (La-DTL) for cross-specialty NER, so that a medical NER system designed for one specialty could be conveniently applied to another one with minimal annotation efforts. The transferability is guaranteed by two components: (i) we propose label-aware MMD for feature representation transfer, and (ii) we perform parameter transfer with a theoretical upper bound which is also label aware. We conduct extensive experiments on 12 cross-specialty NER tasks. The experimental results demonstrate that La-DTL provides consistent accuracy improvement over strong baselines. Besides, the promising experimental results on non-medical NER scenarios indicate that La-DTL is potential to be seamlessly adapted to a wide range of NER tasks.