Dimensionality reduction techniques are widely used for visualizing high-dimensional data in two dimensions. Existing methods are typically designed to preserve either local (e.g., $t$-SNE, UMAP) or global (e.g., MDS, PCA) structure of the data, but none of the established methods can represent both aspects well. In this paper, we present DREAMS (Dimensionality Reduction Enhanced Across Multiple Scales), a method that combines the local structure preservation of $t$-SNE with the global structure preservation of PCA via a simple regularization term. Our approach generates a spectrum of embeddings between the locally well-structured $t$-SNE embedding and the globally well-structured PCA embedding, efficiently balancing both local and global structure preservation. We benchmark DREAMS across eleven real-world datasets, showcasing qualitatively and quantitatively its superior ability to preserve structure across multiple scales compared to previous approaches.

Understanding causal relationships among features is fundamental for explaining machine learning model decisions. However, traditional causal discovery methods face challenges with categorical variables due to numerical instability in conditional independence testing. We propose a dual-encoding causal discovery approach that addresses these limitations by running constraint-based algorithms with complementary encoding strategies and merging results through majority voting. Applied to the Titanic dataset, our method identifies causal structures that align with established explainable methods.

Speech tokenizers are a key building block of fully discrete Speech LLMs. Existing tokenizers either prioritize semantic encoding, fuse semantic content with acoustic style inseparably, or achieve incomplete semantic-acoustic disentanglement. To achieve better disentanglement, we propose DSA-Tokenizer, which explicitly disentangles speech into discrete semantic and acoustic tokens via distinct optimization constraints. Specifically, semantic tokens are supervised by ASR to capture linguistic content, while acoustic tokens focus on mel-spectrograms restoration to encode style. We further introduce a hierarchical Flow Matching decoder and a joint reconstruction-context inpainting training strategy, allowing the model to support both high-fidelity reconstruction and cross-utterance voice clone. To speed up inference, we distill the dit decoder to 4-step inference and improve synthesis quality with GAN fine-tuning. Experiments demonstrate that DSA-Tokenizer provides strong semantic-acoustic disentanglement, reliable controllable voice cloning, and efficient high-fidelity generation with low WER/CER. Moreover, our results suggest that disentangled tokenization provides a more effective interface for downstream large-model speech generation. Audio samples are avaialble at https://anonymous.4open.science/w/DSA_Tokenizer_demo/

Computational cytology faces two major challenges: i) instance-level labels are unreliable and prohibitively costly to obtain, ii) witness rates are extremely low. We propose SLAM-AGS, a Slide-Label-Aware Multitask pretraining framework that jointly optimizes (i) a weakly supervised similarity objective on slide-negative patches and (ii) a self-supervised contrastive objective on slide-positive patches, yielding stronger performance on downstream tasks. To stabilize learning, we apply Adaptive Gradient Surgery to tackle conflicting task gradients and prevent model collapse. We integrate the pretrained encoder into an attention-based Multiple Instance Learning aggregator for bag-level prediction and attention-guided retrieval of the most abnormal instances in a bag. On a publicly available bone-marrow cytology dataset, with simulated witness rates from 10% down to 0.5%, SLAM-AGS improves bag-level F1-Score and Top 400 positive cell retrieval over other pretraining methods, with the largest gains at low witness rates, showing that resolving gradient interference enables stable pretraining and better performance on downstream tasks. To facilitate reproducibility, we share our complete implementation and evaluation framework as open source: https://github.com/Ace95/SLAM-AGS.

We explicitly construct zero loss neural network classifiers. We write the weight matrices and bias vectors in terms of cumulative parameters, which determine truncation maps acting recursively on input space. The configurations for the training data considered are (i) sufficiently small, well separated clusters corresponding to each class, and (ii) equivalence classes which are sequentially linearly separable. In the best case, for $Q$ classes of data in $\mathbb{R}^M$, global minimizers can be described with $Q(M+2)$ parameters.

Robots, as AI with physical instantiation, inhabit our social and physical world, where their actions have both social and physical consequences, posing challenges for researchers when designing social robots. This study starts with a scoping review to identify discussions and potential concerns arising from interactions with robotic systems in the context of healthcare, education, and private homes. Two focus groups of technology ethics experts then validated a comprehensive list of key topics and values in human-robot interaction (HRI) literature in these contexts. These insights were integrated into the HRI Value Compass web tool, to help HRI researchers identify these values in robot design. The tool was evaluated in a pilot study. This work benefits the HRI community by highlighting key concerns in human-robot interactions and providing an instrument to help researchers design robots that align with human values, ensuring future robotic systems adhere to these values in social applications.

Node embeddings are a paradigm in non-parametric graph representation learning, where graph nodes are embedded into a given vector space to enable downstream processing. State-of-the-art node-embedding algorithms, such as DeepWalk and node2vec, are based on random-walk notions of node similarity and on contrastive learning. In this work, we introduce the graph neighbor-embedding (graph NE) framework that directly pulls together embedding vectors of adjacent nodes without relying on any random walks. We show that graph NE strongly outperforms state-of-the-art node-embedding algorithms in terms of local structure preservation. Furthermore, we apply graph NE to the 2D node-embedding problem, obtaining graph t-SNE layouts that also outperform existing graph-layout algorithms.

Dynamic scene rendering and reconstruction play a crucial role in computer vision and augmented reality. Recent methods based on 3D Gaussian Splatting (3DGS), have enabled accurate modeling of dynamic urban scenes, but for urban scenes they require both camera and LiDAR data, ground-truth 3D segmentations and motion data in the form of tracklets or pre-defined object templates such as SMPL. In this work, we explore whether a combination of 2D object agnostic priors in the form of depth and point tracking coupled with a signed distance function (SDF) representation for dynamic objects can be used to relax some of these requirements. We present a novel approach that integrates Signed Distance Functions (SDFs) with 3D Gaussian Splatting (3DGS) to create a more robust object representation by harnessing the strengths of both methods. Our unified optimization framework enhances the geometric accuracy of 3D Gaussian splatting and improves deformation modeling within the SDF, resulting in a more adaptable and precise representation. We demonstrate that our method achieves state-of-the-art performance in rendering metrics even without LiDAR data on urban scenes. When incorporating LiDAR, our approach improved further in reconstructing and generating novel views across diverse object categories, without ground-truth 3D motion annotation. Additionally, our method enables various scene editing tasks, including scene decomposition, and scene composition.

The advent of large language models (LLMs) has revolutionized the integration of knowledge graphs (KGs) in biomedical and cognitive sciences, overcoming limitations in traditional machine learning methods for capturing intricate semantic links among genes, diseases, and cognitive processes. We introduce MultiCNKG, an innovative framework that merges three key knowledge sources: the Cognitive Neuroscience Knowledge Graph (CNKG) with 2.9K nodes and 4.3K edges across 9 node types and 20 edge types; Gene Ontology (GO) featuring 43K nodes and 75K edges in 3 node types and 4 edge types; and Disease Ontology (DO) comprising 11.2K nodes and 8.8K edges with 1 node type and 2 edge types. Leveraging LLMs like GPT-4, we conduct entity alignment, semantic similarity computation, and graph augmentation to create a cohesive KG that interconnects genetic mechanisms, neurological disorders, and cognitive functions. The resulting MultiCNKG encompasses 6.9K nodes across 5 types (e.g., Genes, Diseases, Cognitive Processes) and 11.3K edges spanning 7 types (e.g., Causes, Associated with, Regulates), facilitating a multi-layered view from molecular to behavioral domains. Assessments using metrics such as precision (85.20%), recall (87.30%), coverage (92.18%), graph consistency (82.50%), novelty detection (40.28%), and expert validation (89.50%) affirm its robustness and coherence. Link prediction evaluations with models like TransE (MR: 391, MRR: 0.411) and RotatE (MR: 263, MRR: 0.395) show competitive performance against benchmarks like FB15k-237 and WN18RR. This KG advances applications in personalized medicine, cognitive disorder diagnostics, and hypothesis formulation in cognitive neuroscience.

We present a manifold-based machine learning encoder-decoder method for learning dynamics in time, notably partial differential equations (PDEs), in which the manifold latent space evolves according to Ricci flow. This can be accomplished by parameterizing the latent manifold stage and subsequently simulating Ricci flow in a physics-informed setting, matching manifold quantities so that Ricci flow is empirically achieved. We emphasize dynamics that admit low-dimensional representations. With our method, the manifold, induced by the metric, is discerned through the training procedure, while the latent evolution due to Ricci flow provides an accommodating representation. By use of this flow, we sustain a canonical manifold latent representation for all values in the ambient PDE time interval continuum. We showcase that the Ricci flow facilitates qualities such as learning for out-of-distribution data and adversarial robustness on select PDE data. Moreover, we provide a thorough expansion of our methods in regard to special cases which allow higher-dimensional representations, such as Ricci flow on the hypersphere and neural discovery of non-parametric geometric flows with entropic strategies.

Automating data extraction from full-text randomised controlled trials (RCTs) for meta-analysis remains a significant challenge. This study evaluates the practical performance of three LLMs (Gemini-2.0-flash, Grok-3, GPT-4o-mini) across tasks involving statistical results, risk-of-bias assessments, and study-level characteristics in three medical domains: hypertension, diabetes, and orthopaedics. We tested four distinct prompting strategies (basic prompting, self-reflective prompting, model ensemble, and customised prompts) to determine how to improve extraction quality. All models demonstrate high precision but consistently suffer from poor recall by omitting key information. We found that customised prompts were the most effective, boosting recall by up to 15\%. Based on this analysis, we propose a three-tiered set of guidelines for using LLMs in data extraction, matching data types to appropriate levels of automation based on task complexity and risk. Our study offers practical advice for automating data extraction in real-world meta-analyses, balancing LLM efficiency with expert oversight through targeted, task-specific automation.

The rapid advancement of speech generation models has heightened privacy and security concerns related to voice cloning (VC). Recent studies have investigated disrupting unauthorized voice cloning by introducing adversarial perturbations. However, determined attackers can mitigate these protective perturbations and successfully execute VC. In this study, we conduct the first systematic evaluation of these protective perturbations against VC under realistic threat models that include perturbation purification. Our findings reveal that while existing purification methods can neutralize a considerable portion of the protective perturbations, they still lead to distortions in the feature space of VC models, which degrades the performance of VC. From this perspective, we propose a novel two-stage purification method: (1) Purify the perturbed speech; (2) Refine it using phoneme guidance to align it with the clean speech distribution. Experimental results demonstrate that our method outperforms state-of-the-art purification methods in disrupting VC defenses. Our study reveals the limitations of adversarial perturbation-based VC defenses and underscores the urgent need for more robust solutions to mitigate the security and privacy risks posed by VC. The code and audio samples are available at https://de-antifake.github.io.

Self-supervised learning (SSL) allows training data representations without a supervised signal and has become an important paradigm in machine learning. Most SSL methods employ the cosine similarity between embedding vectors and hence effectively embed data on a hypersphere. While this seemingly implies that embedding norms cannot play any role in SSL, a few recent works have suggested that embedding norms have properties related to network convergence and confidence. In this paper, we resolve this apparent contradiction and systematically establish the embedding norm's role in SSL training. Using theoretical analysis, simulations, and experiments, we show that embedding norms (i) govern SSL convergence rates and (ii) encode network confidence, with smaller norms corresponding to unexpected samples. Additionally, we show that manipulating embedding norms can have large effects on convergence speed. Our findings demonstrate that SSL embedding norms are integral to understanding and optimizing network behavior.

We analyze geometric aspects of the gradient descent algorithm in Deep Learning (DL), and give a detailed discussion of the circumstance that in underparametrized DL networks, zero loss minimization can generically not be attained. As a consequence, we conclude that the distribution of training inputs must necessarily be non-generic in order to produce zero loss minimizers, both for the method constructed in [Chen-Munoz Ewald 2023, 2024], or for gradient descent [Chen 2025] (which assume clustering of training data).

Temporal claims, often riddled with inaccuracies, are a significant challenge in the digital misinformation landscape. Fact-checking systems that can accurately verify such claims are crucial for combating misinformation. Current systems struggle with the complexities of evaluating the accuracy of these claims, especially when they include multiple, overlapping, or recurring events. We introduce a novel timeline-based fact verification framework that identify events from both claim and evidence and organize them into their respective chronological timelines. The framework systematically examines the relationships between the events in both claim and evidence to predict the veracity of each claim event and their chronological accuracy. This allows us to accurately determine the overall veracity of the claim. We also introduce a new dataset of complex temporal claims involving timeline-based reasoning for the training and evaluation of our proposed framework. Experimental results demonstrate the effectiveness of our approach in handling the intricacies of temporal claim verification.

Thompson sampling has proven effective across a wide range of stationary bandit environments. However, as we demonstrate in this paper, it can perform poorly when applied to non-stationary environments. We show that such failures are attributed to the fact that, when exploring, the algorithm does not differentiate actions based on how quickly the information acquired loses its usefulness due to nonstationarity. Building upon this insight, we propose predictive sampling, an algorithm that deprioritizes acquiring information that quickly loses usefulness. Theoretical guarantee on the performance of predictive sampling is established through a Bayesian regret bound. We provide versions of predictive sampling for which computations tractably scale to complex bandit environments of practical interest. Through numerical simulations, we demonstrate that predictive sampling outperforms Thompson sampling in all non-stationary environments examined.

In the rapidly evolving landscape of human-computer interaction, the integration of vision capabilities into conversational agents stands as a crucial advancement. This paper presents an initial implementation of a dialogue manager that leverages the latest progress in Large Language Models (e.g., GPT-4, IDEFICS) to enhance the traditional text-based prompts with real-time visual input. LLMs are used to interpret both textual prompts and visual stimuli, creating a more contextually aware conversational agent. The system's prompt engineering, incorporating dialogue with summarisation of the images, ensures a balance between context preservation and computational efficiency. Six interactions with a Furhat robot powered by this system are reported, illustrating and discussing the results obtained. By implementing this vision-enabled dialogue system, the paper envisions a future where conversational agents seamlessly blend textual and visual modalities, enabling richer, more context-aware dialogues.

Machine Learning and AI have the potential to transform data-driven scientific discovery, enabling accurate predictions for several scientific phenomena. As many scientific questions are inherently causal, this paper looks at the causal inference task of treatment effect estimation, where the outcome of interest is recorded in high-dimensional observations in a Randomized Controlled Trial (RCT). Despite being the simplest possible causal setting and a perfect fit for deep learning, we theoretically find that many common choices in the literature may lead to biased estimates. To test the practical impact of these considerations, we recorded ISTAnt, the first real-world benchmark for causal inference downstream tasks on high-dimensional observations as an RCT studying how garden ants (Lasius neglectus) respond to microparticles applied onto their colony members by hygienic grooming. Comparing 6 480 models fine-tuned from state-of-the-art visual backbones, we find that the sampling and modeling choices significantly affect the accuracy of the causal estimate, and that classification accuracy is not a proxy thereof. We further validated the analysis, repeating it on a synthetically generated visual data set controlling the causal model. Our results suggest that future benchmarks should carefully consider real downstream scientific questions, especially causal ones. Further, we highlight guidelines for representation learning methods to help answer causal questions in the sciences.

Survival prognosis is crucial for medical informatics. Practitioners often confront small-sized clinical data, especially cancer patient cases, which can be insufficient to induce useful patterns for survival predictions. This study deals with small sample survival analysis by leveraging transfer learning, a useful machine learning technique that can enhance the target analysis with related knowledge pre-learned from other data. We propose and develop various transfer learning methods designed for common survival models. For parametric models such as DeepSurv, Cox-CC (Cox-based neural networks), and DeepHit (end-to-end deep learning model), we apply standard transfer learning techniques like pretraining and fine-tuning. For non-parametric models such as Random Survival Forest, we propose a new transfer survival forest (TSF) model that transfers tree structures from source tasks and fine-tunes them with target data. We evaluated the transfer learning methods on colorectal cancer (CRC) prognosis. The source data are 27,379 SEER CRC stage I patients, and the target data are 728 CRC stage I patients from the West China Hospital. When enhanced by transfer learning, Cox-CC's $C^{td}$ value was boosted from 0.7868 to 0.8111, DeepHit's from 0.8085 to 0.8135, DeepSurv's from 0.7722 to 0.8043, and RSF's from 0.7940 to 0.8297 (the highest performance). All models trained with data as small as 50 demonstrated even more significant improvement. Conclusions: Therefore, the current survival models used for cancer prognosis can be enhanced and improved by properly designed transfer learning techniques. The source code used in this study is available at https://github.com/YonghaoZhao722/TSF.

Federated Learning (FL) has emerged as a promising method to collaboratively learn from decentralized and heterogeneous data available at different clients without the requirement of data ever leaving the clients. Recent works on FL have advocated taking a Bayesian approach to FL as it offers a principled way to account for the model and predictive uncertainty by learning a posterior distribution for the client and/or server models. Moreover, Bayesian FL also naturally enables personalization in FL to handle data heterogeneity across the different clients by having each client learn its own distinct personalized model. In particular, the hierarchical Bayesian approach enables all the clients to learn their personalized models while also taking into account the commonalities via a prior distribution provided by the server. However, despite their promise, Bayesian approaches for FL can be computationally expensive and can have high communication costs as well because of the requirement of computing and sending the posterior distributions. We present a novel Bayesian FL method using an efficient second-order optimization approach, with a computational cost that is similar to first-order optimization methods like Adam, but also provides the various benefits of the Bayesian approach for FL (e.g., uncertainty, personalization), while also being significantly more efficient and accurate than SOTA Bayesian FL methods (both for standard as well as personalized FL settings). Our method achieves improved predictive accuracies as well as better uncertainty estimates as compared to the baselines which include both optimization based as well as Bayesian FL methods.

Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much higher dimensionality. We show that in this case traditional persistent homology becomes very sensitive to noise and fails to detect the correct topology. The same holds true for existing refinements of persistent homology. As a remedy, we find that spectral distances on the k-nearest-neighbor graph of the data, such as diffusion distance and effective resistance, allow to detect the correct topology even in the presence of high-dimensional noise. Moreover, we derive a novel closed-form formula for effective resistance, and describe its relation to diffusion distances. Finally, we apply these methods to high-dimensional single-cell RNA-sequencing data and show that spectral distances allow robust detection of cell cycle loops.

Warning: Contents of this paper may be upsetting. Public attitudes towards key societal issues, expressed on online media, are of immense value in policy and reform efforts, yet challenging to understand at scale. We study one such social issue: homelessness in the U.S., by leveraging the remarkable capabilities of large language models to assist social work experts in analyzing millions of posts from Twitter. We introduce a framing typology: Online Attitudes Towards Homelessness (OATH) Frames: nine hierarchical frames capturing critiques, responses and perceptions. We release annotations with varying degrees of assistance from language models, with immense benefits in scaling: 6.5x speedup in annotation time while only incurring a 3 point F1 reduction in performance with respect to the domain experts. Our experiments demonstrate the value of modeling OATH-Frames over existing sentiment and toxicity classifiers. Our large-scale analysis with predicted OATH-Frames on 2.4M posts on homelessness reveal key trends in attitudes across states, time periods and vulnerable populations, enabling new insights on the issue. Our work provides a general framework to understand nuanced public attitudes at scale, on issues beyond homelessness.

Representing systems using the SAPPhIRE causality model is found useful in supporting design-by-analogy. However, creating a SAPPhIRE model of artificial or biological systems is an effort-intensive process that requires human experts to source technical knowledge from multiple technical documents regarding how the system works. This research investigates how to leverage Large Language Models (LLMs) in creating structured descriptions of systems using the SAPPhIRE model of causality. This paper, the second part of the two-part research, presents a new Retrieval-Augmented Generation (RAG) tool for generating information related to SAPPhIRE constructs of artificial systems and reports the results from a preliminary evaluation of the tool's success - focusing on the factual accuracy and reliability of outcomes.

In this paper, we present a new solution to the problem of large-scale multi-building and multi-floor indoor localization based on linked neural networks, where each neural network is dedicated to a sub-problem and trained under a hierarchical stage-wise training framework. When the measured data from sensors have a hierarchical representation as in multi-building and multi-floor indoor localization, it is important to exploit the hierarchical nature in data processing to provide a scalable solution. In this regard, the hierarchical stage-wise training framework extends the original stage-wise training framework to the case of multiple linked networks by training a lower-hierarchy network based on the prior knowledge gained from the training of higher-hierarchy networks. The experimental results with the publicly-available UJIIndoorLoc multi-building and multi-floor Wi-Fi RSSI fingerprint database demonstrate that the linked neural networks trained under the proposed hierarchical stage-wise training framework can achieve a three-dimensional localization error of 8.19 m, which, to the best of the authors' knowledge, is the most accurate result ever obtained for neural network-based models trained and evaluated with the full datasets of the UJIIndoorLoc database, and that, when applied to a model based on hierarchical convolutional neural networks, the proposed training framework can also significantly reduce the three-dimensional localization error from 11.78 m to 8.71 m.

We propose a method for the classification of objects that are structured as random trees. Our aim is to model a distribution over the node label assignments in settings where the tree data structure is associated with node attributes (typically high dimensional embeddings). The tree topology is not predetermined and none of the label assignments are present during inference. Other methods that produce a distribution over node label assignment in trees (or more generally in graphs) either assume conditional independence of the label assignment, operate on a fixed graph topology, or require part of the node labels to be observed. Our method defines a Markov Network with the corresponding topology of the random tree and an associated Gibbs distribution. We parameterize the Gibbs distribution with a Graph Neural Network that operates on the random tree and the node embeddings. This allows us to estimate the likelihood of node assignments for a given random tree and use MCMC to sample from the distribution of node assignments. We evaluate our method on the tasks of node classification in trees on the Stanford Sentiment Treebank dataset. Our method outperforms the baselines on this dataset, demonstrating its effectiveness for modeling joint distributions of node labels in random trees.

Representation of systems using the SAPPhIRE model of causality can be an inspirational stimulus in design. However, creating a SAPPhIRE model of a technical or a natural system requires sourcing technical knowledge from multiple technical documents regarding how the system works. This research investigates how to generate technical content accurately relevant to the SAPPhIRE model of causality using a Large Language Model, also called LLM. This paper, which is the first part of the two-part research, presents a method for hallucination suppression using Retrieval Augmented Generating with LLM to generate technical content supported by the scientific information relevant to a SAPPhIRE con-struct. The result from this research shows that the selection of reference knowledge used in providing context to the LLM for generating the technical content is very important. The outcome of this research is used to build a software support tool to generate the SAPPhIRE model of a given technical system.

Do large language models (LLMs) display rational reasoning? LLMs have been shown to contain human biases due to the data they have been trained on; whether this is reflected in rational reasoning remains less clear. In this paper, we answer this question by evaluating seven language models using tasks from the cognitive psychology literature. We find that, like humans, LLMs display irrationality in these tasks. However, the way this irrationality is displayed does not reflect that shown by humans. When incorrect answers are given by LLMs to these tasks, they are often incorrect in ways that differ from human-like biases. On top of this, the LLMs reveal an additional layer of irrationality in the significant inconsistency of the responses. Aside from the experimental results, this paper seeks to make a methodological contribution by showing how we can assess and compare different capabilities of these types of models, in this case with respect to rational reasoning.

Planning under uncertainty is critical to robotics. The Partially Observable Markov Decision Process (POMDP) is a mathematical framework for such planning problems. It is powerful due to its careful quantification of the non-deterministic effects of actions and partial observability of the states. But precisely because of this, POMDP is notorious for its high computational complexity and deemed impractical for robotics. However, since early 2000, POMDPs solving capabilities have advanced tremendously, thanks to sampling-based approximate solvers. Although these solvers do not generate the optimal solution, they can compute good POMDP solutions that significantly improve the robustness of robotics systems within reasonable computational resources, thereby making POMDPs practical for many realistic robotics problems. This paper presents a review of POMDPs, emphasizing computational issues that have hindered its practicality in robotics and ideas in sampling-based solvers that have alleviated such difficulties, together with lessons learned from applying POMDPs to physical robots.

To date, the privacy-protection intended pixelation tasks are still labor-intensive and yet to be studied. With the prevailing of video live streaming, establishing an online face pixelation mechanism during streaming is an urgency. In this paper, we develop a new method called Face Pixelation in Video Live Streaming (FPVLS) to generate automatic personal privacy filtering during unconstrained streaming activities. Simply applying multi-face trackers will encounter problems in target drifting, computing efficiency, and over-pixelation. Therefore, for fast and accurate pixelation of irrelevant people's faces, FPVLS is organized in a frame-to-video structure of two core stages. On individual frames, FPVLS utilizes image-based face detection and embedding networks to yield face vectors. In the raw trajectories generation stage, the proposed Positioned Incremental Affinity Propagation (PIAP) clustering algorithm leverages face vectors and positioned information to quickly associate the same person's faces across frames. Such frame-wise accumulated raw trajectories are likely to be intermittent and unreliable on video level. Hence, we further introduce the trajectory refinement stage that merges a proposal network with the two-sample test based on the Empirical Likelihood Ratio (ELR) statistic to refine the raw trajectories. A Gaussian filter is laid on the refined trajectories for final pixelation. On the video live streaming dataset we collected, FPVLS obtains satisfying accuracy, real-time efficiency, and contains the over-pixelation problems.

We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem (UAT) for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.