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This paper studies the problem of forecasting general stochastic processes using a path-dependent extension of the Neural Jump ODE (NJ-ODE) framework \citep{herrera2021neural}. While NJ-ODE was the first framework to establish convergence guarantees for the prediction of irregularly observed time series, these results were limited to data stemming from It\^o-diffusions with complete observations, in particular Markov processes, where all coordinates are observed simultaneously. In this work, we generalise these results to generic, possibly non-Markovian or discontinuous, stochastic processes with incomplete observations, by utilising the reconstruction properties of the signature transform. These theoretical results are supported by empirical studies, where it is shown that the path-dependent NJ-ODE outperforms the original NJ-ODE framework in the case of non-Markovian data. Moreover, we show that PD-NJ-ODE can be applied successfully to classical stochastic filtering problems and to limit order book (LOB) data.

We investigate two efficient time discretizations for the post-processing technique of discontinuous Galerkin (DG) methods to solve hyperbolic conservation laws. The post-processing technique, which is applied at the final time of the DG method, can enhance the accuracy of the original DG solution (spatial superconvergence). One main difficulty of the post-processing technique is that the spatial superconvergence after post-processing needs to be matched with proper temporary accuracy. If the semi-discretized system (ODE system after spatial discretization) is under-resolved in time, then the space superconvergence will be concealed. In this paper, we focus our investigation on the recently designed SDG method and derive its explicit scheme from a correction process based on the DG weak formulation. We also introduce another similar technique, namely the spectral deferred correction (SDC) method. A comparison is made among both proposed time discretization techniques with the standard third-order Runge-Kutta method through several numerical examples, to conclude that both the SDG and SDC methods are efficient time discretization techniques for exploiting the spatial superconvergence of the DG methods.

These are self-contained lecture notes for spectral independence. For an $n$-vertex graph, the spectral independence condition is a bound on the maximum eigenvalue of the $n\times n$ influence matrix whose entries capture the influence between pairs of vertices, it is closely related to the covariance matrix. We will present recent results showing that spectral independence implies the mixing time of the Glauber dynamics is polynomial (where the degree of the polynomial depends on certain parameters). The proof utilizes local-to-global theorems which we will detail in these notes. Finally, we will present more recent results showing that spectral independence implies an optimal bound on the relaxation time (inverse spectral gap) and with some additional conditions implies an optimal mixing time bound of $O(n\log{n})$ for the Glauber dynamics. We also present the results of Anari, Liu, Oveis Gharan, and Vinzant (2019) for generating a random basis of a matroid. The analysis of the associated bases-exchange walk utilizes the local-to-global theorems used for spectral independence with the Trickle-Down Theorem of Oppenheim (2018) to analyze the local walks. Our focus in these notes is on the analysis of the spectral gap of the associated Markov chains from a functional analysis perspective, and we present proofs of the associated local-to-global theorems from this same Markov chain perspective.

Sound event localization and detection (SELD) combines two subtasks: sound event detection (SED) and direction of arrival (DOA) estimation. SELD is usually tackled as an audio-only problem, but visual information has been recently included. Few audio-visual (AV)-SELD works have been published and most employ vision via face/object bounding boxes, or human pose keypoints. In contrast, we explore the integration of audio and visual feature embeddings extracted with pre-trained deep networks. For the visual modality, we tested ResNet50 and Inflated 3D ConvNet (I3D). Our comparison of AV fusion methods includes the AV-Conformer and Cross-Modal Attentive Fusion (CMAF) model. Our best models outperform the DCASE 2023 Task3 audio-only and AV baselines by a wide margin on the development set of the STARSS23 dataset, making them competitive amongst state-of-the-art results of the AV challenge, without model ensembling, heavy data augmentation, or prediction post-processing. Such techniques and further pre-training could be applied as next steps to improve performance.

The learning dynamics of deep neural networks are not well understood. The information bottleneck (IB) theory proclaimed separate fitting and compression phases. But they have since been heavily debated. We comprehensively analyze the learning dynamics by investigating a layer's reconstruction ability of the input and prediction performance based on the evolution of parameters during training. We empirically show the existence of three phases using common datasets and architectures such as ResNet and VGG: (i) near constant reconstruction loss, (ii) decrease, and (iii) increase. We also derive an empirically grounded data model and prove the existence of phases for single-layer networks. Technically, our approach leverages classical complexity analysis. It differs from IB by relying on measuring reconstruction loss rather than information theoretic measures to relate information of intermediate layers and inputs. Our work implies a new best practice for transfer learning: We show empirically that the pre-training of a classifier should stop well before its performance is optimal.

Digital quantum simulation has broad applications in approximating unitary evolutions of Hamiltonians. In practice, many simulation tasks for quantum systems focus on quantum states in the low-energy subspace instead of the entire Hilbert space. In this paper, we systematically investigate the complexity of digital quantum simulation based on product formulas in the low-energy subspace. We show that the simulation error depends on the effective low-energy norm of the Hamiltonian for a variety of digital quantum simulation algorithms and quantum systems, allowing improvements over the previous complexities for full unitary simulations even for imperfect state preparations. In particular, for simulating spin models in the low-energy subspace, we prove that randomized product formulas such as qDRIFT and random permutation require smaller step complexities. This improvement also persists in symmetry-protected digital quantum simulations. We prove a similar improvement in simulating the dynamics of power-law quantum interactions. We also provide a query lower bound for general digital quantum simulations in the low-energy subspace.

We present a novel method, the Chain of Empathy (CoE) prompting, that utilizes insights from psychotherapy to induce Large Language Models (LLMs) to reason about human emotional states. This method is inspired by various psychotherapy approaches including Cognitive Behavioral Therapy (CBT), Dialectical Behavior Therapy (DBT), Person Centered Therapy (PCT), and Reality Therapy (RT), each leading to different patterns of interpreting clients' mental states. LLMs without reasoning generated predominantly exploratory responses. However, when LLMs used CoE reasoning, we found a more comprehensive range of empathetic responses aligned with the different reasoning patterns of each psychotherapy model. The CBT based CoE resulted in the most balanced generation of empathetic responses. The findings underscore the importance of understanding the emotional context and how it affects human and AI communication. Our research contributes to understanding how psychotherapeutic models can be incorporated into LLMs, facilitating the development of context-specific, safer, and empathetic AI.

The advent of large language models marks a revolutionary breakthrough in artificial intelligence. With the unprecedented scale of training and model parameters, the capability of large language models has been dramatically improved, leading to human-like performances in understanding, language synthesizing, and common-sense reasoning, etc. Such a major leap-forward in general AI capacity will change the pattern of how personalization is conducted. For one thing, it will reform the way of interaction between humans and personalization systems. Instead of being a passive medium of information filtering, large language models present the foundation for active user engagement. On top of such a new foundation, user requests can be proactively explored, and user's required information can be delivered in a natural and explainable way. For another thing, it will also considerably expand the scope of personalization, making it grow from the sole function of collecting personalized information to the compound function of providing personalized services. By leveraging large language models as general-purpose interface, the personalization systems may compile user requests into plans, calls the functions of external tools to execute the plans, and integrate the tools' outputs to complete the end-to-end personalization tasks. Today, large language models are still being developed, whereas the application in personalization is largely unexplored. Therefore, we consider it to be the right time to review the challenges in personalization and the opportunities to address them with LLMs. In particular, we dedicate this perspective paper to the discussion of the following aspects: the development and challenges for the existing personalization system, the newly emerged capabilities of large language models, and the potential ways of making use of large language models for personalization.

We introduce a multi-task setup of identifying and classifying entities, relations, and coreference clusters in scientific articles. We create SciERC, a dataset that includes annotations for all three tasks and develop a unified framework called Scientific Information Extractor (SciIE) for with shared span representations. The multi-task setup reduces cascading errors between tasks and leverages cross-sentence relations through coreference links. Experiments show that our multi-task model outperforms previous models in scientific information extraction without using any domain-specific features. We further show that the framework supports construction of a scientific knowledge graph, which we use to analyze information in scientific literature.

Within the rapidly developing Internet of Things (IoT), numerous and diverse physical devices, Edge devices, Cloud infrastructure, and their quality of service requirements (QoS), need to be represented within a unified specification in order to enable rapid IoT application development, monitoring, and dynamic reconfiguration. But heterogeneities among different configuration knowledge representation models pose limitations for acquisition, discovery and curation of configuration knowledge for coordinated IoT applications. This paper proposes a unified data model to represent IoT resource configuration knowledge artifacts. It also proposes IoT-CANE (Context-Aware recommendatioN systEm) to facilitate incremental knowledge acquisition and declarative context driven knowledge recommendation.