Modeling and reconstructing multidimensional physical dynamics from sparse and off-grid observations presents a fundamental challenge in scientific research. Recently, diffusion-based generative modeling shows promising potential for physical simulation. However, current approaches typically operate on on-grid data with preset spatiotemporal resolution, but struggle with the sparsely observed and continuous nature of real-world physical dynamics. To fill the gaps, we present SDIFT, Sequential DIffusion in Functional Tucker space, a novel framework that generates full-field evolution of physical dynamics from irregular sparse observations. SDIFT leverages the functional Tucker model as the latent space representer with proven universal approximation property, and represents observations as latent functions and Tucker core sequences. We then construct a sequential diffusion model with temporally augmented UNet in the functional Tucker space, denoising noise drawn from a Gaussian process to generate the sequence of core tensors. At the posterior sampling stage, we propose a Message-Passing Posterior Sampling mechanism, enabling conditional generation of the entire sequence guided by observations at limited time steps. We validate SDIFT on three physical systems spanning astronomical (supernova explosions, light-year scale), environmental (ocean sound speed fields, kilometer scale), and molecular (organic liquid, millimeter scale) domains, demonstrating significant improvements in both reconstruction accuracy and computational efficiency compared to state-of-the-art approaches.

Large language models (LLMs) have shown great potential as general-purpose AI assistants across various domains. To fully leverage this potential in specific applications, many companies provide fine-tuning API services, enabling users to upload their own data for LLM customization. However, fine-tuning services introduce a new safety threat: user-uploaded data, whether harmful or benign, can break the model's alignment, leading to unsafe outputs. Moreover, existing defense methods struggle to address the diversity of fine-tuning datasets (e.g., varying sizes, tasks), often sacrificing utility for safety or vice versa. To address this issue, we propose Safe Delta, a safety-aware post-training defense method that adjusts the delta parameters (i.e., the parameter change before and after fine-tuning). Specifically, Safe Delta estimates the safety degradation, selects delta parameters to maximize utility while limiting overall safety loss, and applies a safety compensation vector to mitigate residual safety loss. Through extensive experiments on four diverse datasets with varying settings, our approach consistently preserves safety while ensuring that the utility gain from benign datasets remains unaffected.

Foundation models have demonstrated remarkable performance across modalities such as language and vision. However, model reuse across distinct modalities (e.g., text and vision) remains limited due to the difficulty of aligning internal representations. Existing methods require extensive paired training data or are constrained to specific domains. We introduce a semi-supervised approach for model alignment via conditional flow matching. The conditional flow between latent spaces of different modalities (e.g., text-to-image or biological-to-artificial neuronal activity) can be learned in two settings: ($1$) solving a (balanced or unbalanced) optimal transport problem with an inter-space bridge cost, and ($2$) performing memory-efficient alignment using labelled exemplars. Despite being constrained by the original models' capacity, our method--under both settings--matches downstream task performance of end-to-end trained models on object recognition and image generation tasks across MNIST, ImageNet, and \cite{majaj2015simple} datasets, particularly when labelled training data is scarce ($<20\%$). Our method provides a data-efficient solution for inter-modal model alignment with minimal supervision.

Variational Autoencoders (VAEs) are well-established as a principled approach to probabilistic unsupervised learning with neural networks. Typically, an encoder network defines the parameters of a Gaussian distributed latent space from which we can sample and pass realizations to a decoder network. This model is trained to reconstruct its inputs and is optimized through the evidence lower bound. In recent years, discrete latent spaces have grown in popularity, suggesting that they may be a natural choice for many data modalities (e.g. text). In this tutorial, we provide a rigorous, yet practical, introduction to discrete variational autoencoders -- specifically, VAEs in which the latent space is made up of latent variables that follow a categorical distribution. We assume only a basic mathematical background with which we carefully derive each step from first principles. From there, we develop a concrete training recipe and provide an example implementation, hosted at https://github.com/alanjeffares/discreteVAE.

In this technical note, we introduce and analyze AWNN: an adaptively weighted nearest neighbor method for performing matrix completion. Nearest neighbor (NN) methods are widely used in missing data problems across multiple disciplines such as in recommender systems and for performing counterfactual inference in panel data settings. Prior works have shown that in addition to being very intuitive and easy to implement, NN methods enjoy nice theoretical guarantees. However, the performance of majority of the NN methods rely on the appropriate choice of the radii and the weights assigned to each member in the nearest neighbor set and despite several works on nearest neighbor methods in the past two decades, there does not exist a systematic approach of choosing the radii and the weights without relying on methods like cross-validation. AWNN addresses this challenge by judiciously balancing the bias variance trade off inherent in weighted nearest-neighbor regression. We provide theoretical guarantees for the proposed method under minimal assumptions and support the theory via synthetic experiments.

--Downstream probing has been the dominant method for evaluating model representations, an important process given the increasing prominence of self-supervised learning and foundation models. However, downstream probing primarily assesses the availability of task-relevant information in the model's latent space, overlooking attributes such as equivariance, invariance, and disentanglement, which contribute to the interpretability, adaptability, and utility of representations in real-world applications. While some attempts have been made to measure these qualities in representations, no unified evaluation framework with modular, generalizable, and interpretable metrics exists. In this paper, we argue for the importance of representation evaluation beyond downstream probing. We introduce a standardized protocol to quantify informativeness, equivariance, invariance, and disentanglement of factors of variation in model representations. We use it to evaluate representations from a variety of models in the image and speech domains using different architectures and pretraining approaches on identified controllable factors of variation. We find that representations from models with similar downstream performance can behave substantially differently with regard to these attributes. This hints that the respective mechanisms underlying their downstream performance are functionally different, prompting new research directions to understand and improve representations. Representation learning has become popular across many fields due to its effectiveness, computational efficiency, and the relative simplicity of using representations from pretrained models as features for various downstream tasks. Many architectures, training paradigms, and modalities have been used to learn representations that are effective in a variety of tasks, such as retrieval, classification, and generation.

--Cyber-Physical Systems (CPS) are abundant in safety-critical domains such as healthcare, avionics, and autonomous vehicles. Formal verification of their operational safety is, therefore, of utmost importance. In this paper, we address the falsification problem, where the focus is on searching for an unsafe execution in the system instead of proving their absence. The contribution of this paper is a framework that (a) connects the falsification of CPS with the falsification of deep neural networks (DNNs) and (b) leverages the inherent interpretability of Decision Trees for faster falsification of CPS. This is achieved by: (1) building a surrogate model of the CPS under test, either as a DNN model or a Decision Tree, (2) application of various DNN falsification tools to falsify CPS, and (3) a novel falsification algorithm guided by the explanations of safety violations of the CPS model extracted from its Decision Tree surrogate. The proposed framework has the potential to exploit a repertoire of adversarial attack algorithms designed to falsify robustness properties of DNNs, as well as state-of-the-art falsification algorithms for DNNs. Although the presented methodology is applicable to systems that can be executed/simulated in general, we demonstrate its effectiveness, particularly in CPS. Decision tree-guided falsification shows promising results in efficiently finding multiple counterexamples in the ARCH-COMP 2024 falsification benchmarks [22]. The traditional simulation and testing techniques can be effective for debugging the early stages of Cyber-Physical-Systems (CPS) design. However, as the design becomes pristine by passing through multiple phases of testing, finding the lurking bugs becomes computationally expensive and challenging by means of simulation and testing alone. Formal verification techniques such as model-checking come in handy here by either proving the absence of bugs in such designs or by providing a counterexample behavior that violates the specification. A complementary approach is falsification, where the focus is solely on discovering a system behavior that is a counterexample to a given specification. In this work, we address the falsification of safety specifications expressed in signal temporal logic [27] for CPS given as an executable. Our Contribution The contribution of this paper is a falsification framework that employs two strategies. First, it connects the falsification of reachability specifications of CPS with the falsification of reachability specifications of deep neural networks (DNNs). A. Kundu and S. Gon are students of the Indian Association for the Cultivation of Science (IACS), India.

In this note research and publications by Zdzisław Pawlak and his collaborators from 1970s and 1981 are recalled. Focus is placed on the sources of inspiration which one can identify on the basis of those publications. Finally, developments from 1981 related to rough sets and information systems are outlined.

Despite the empirical success of foundation models, we do not have a systematic characterization of the representations that these models learn. In this paper, we establish the contexture theory. It shows that a large class of representation learning methods can be characterized as learning from the association between the input and a context variable. Specifically, we show that many popular methods aim to approximate the top-d singular functions of the expectation operator induced by the context, in which case we say that the representation learns the contexture. We demonstrate the generality of the contexture theory by proving that representation learning within various learning paradigms -- supervised, self-supervised, and manifold learning -- can all be studied from such a perspective. We also prove that the representations that learn the contexture are optimal on those tasks that are compatible with the context. One important implication of the contexture theory is that once the model is large enough to approximate the top singular functions, further scaling up the model size yields diminishing returns. Therefore, scaling is not all we need, and further improvement requires better contexts. To this end, we study how to evaluate the usefulness of a context without knowing the downstream tasks. We propose a metric and show by experiments that it correlates well with the actual performance of the encoder on many real datasets.

In particular, Theorem 5 applies when null A δ = { δ }, i.e., when θ null A is just a standard, non-robust, solution. This is different from [56], whose main result is only applicable to solutions satisfying the infinitely many constraints f (θ, δ) 0, δ A δ i, i = 1,...,N, where A δ i is tuned to the Wasserstein bound. As previously noted, R plays the role of a tunable parameter, and the result in Theorem 5 holds for any choice of the value ofR . As a consequence, the user can play with R to optimize the bound on Risk ( θ null A) given in Theorem 5. As R increases, s A, null A (and, thereby, ε (s A, null A)) tends to increase while µ/R diminishes. While the best compromise is difficult to foresee, one can experimentally try various choices R 1 < R 2 < < R i < R h and select the one giving the best result. The corresponding confidence level can be bounded as follows: P Nnull D: Risk (θ null A) > ε (s A, null A,i) + µ R i for at least one i { 1,...h } null h null i =1P Nnull D: Risk (θ null A) > ε (s A, null A,i) + µ R i null h null i =1β = hβ, 29 from which P Nnull D: Risk ( θ null A) ε ( s A, null A,i) + µ R i for all i = 1,...h null 1 hβ.