For all datasets, we used standard normalization that scales the features to have zero mean and standard deviation of one. The architecture of the autoencoder consists of one hidden layer with sigmoid activation. A linear activation is used for the output layer. We use a hidden layer of 200 neurons for all datasets. We trained each dataset for 10 epochs using stochastic gradient descent with a momentum of 0.9 and a batch size of 128.

In this paper we introduce the pure exploration transductive linear bandit problem: given a set of measurement vectors $\mathcal{X}\subset \mathbb{R}^d$, a set of items $\mathcal{Z}\subset \mathbb{R}^d$, a fixed confidence $\delta$, and an unknown vector $\theta^{\ast}\in \mathbb{R}^d$, the goal is to infer $\arg\max_{z\in \mathcal{Z}} z^\top\theta^\ast$ with probability $1-\delta$ by making as few sequentially chosen noisy measurements of the form $x^\top\theta^{\ast}$ as possible. When $\mathcal{X}=\mathcal{Z}$, this setting generalizes linear bandits, and when $\mathcal{X}$ is the standard basis vectors and $\mathcal{Z}\subset \{0,1\}^d$, combinatorial bandits. The transductive setting naturally arises when the set of measurement vectors is limited due to factors such as availability or cost. As an example, in drug discovery the compounds and dosages $\mathcal{X}$ a practitioner may be willing to evaluate in the lab in vitro due to cost or safety reasons may differ vastly from those compounds and dosages $\mathcal{Z}$ that can be safely administered to patients in vivo. Alternatively, in recommender systems for books, the set of books $\mathcal{X}$ a user is queried about may be restricted to known best-sellers even though the goal might be to recommend more esoteric titles $\mathcal{Z}$. In this paper, we provide instance-dependent lower bounds for the transductive setting, an algorithm that matches these up to logarithmic factors, and an evaluation. In particular, we present the first non-asymptotic algorithm for linear bandits that nearly achieves the information-theoretic lower bound.

The main challenges in deriving the convergence rate of the MLE mainly come from two issues: (1) The interaction between the function $h_{0}$ and the density function $f$; (2) The deviated proportion $\lambda^{\ast}$ can go to the extreme points of $[0,1]$ as the sample size tends to infinity. To address these challenges, we develop the \emph{distinguishability condition} to capture the linear independent relation between the function $h_{0}$ and the density function $f$. We then provide comprehensive convergence rates of the MLE via the vanishing rate of $\lambda^{\ast}$ to zero as well as the distinguishability of two functions $h_{0}$ and $f$.

Many networked systems such as electric networks, the brain, and social networks of opinion dynamics are known to obey conservation laws. Examples of this phenomenon include the Kirchoff laws in electric networks and opinion consensus in social networks. Conservation laws in networked systems are modeled as balance equations of the form $X = B^\ast Y$, where the sparsity pattern of $B^\ast \in \mathbb{R}^{p\times p}$ captures the connectivity of the network on $p$ nodes, and $Y, X \in \mathbb{R}^p$ are vectors of ''potentials'' and ''injected flows'' at the nodes respectively. The node potentials $Y$ cause flows across edges which aim to balance out the potential difference, and the flows $X$ injected at the nodes are extraneous to the network dynamics. In several practical systems, the network structure is often unknown and needs to be estimated from data to facilitate modeling, management, and control.

Objective speech quality assessment is central to telephony, V oIP, and streaming systems, where large volumes of degraded audio must be monitored and optimized at scale. Classical metrics such as PESQ and POLQA approximate human mean opinion scores (MOS) but require carefully controlled conditions and expensive listening tests, while learning-based models such as NISQA regress MOS and multiple perceptual dimensions from waveforms or spectrograms, achieving high correlation with subjective ratings yet remaining rigid: they yield fixed scalar scores, do not support interactive, natural-language queries, and do not natively provide textual rationales. In this work, we introduce SpeechQualityLLM, a multimodal speech quality question-answering (QA) system that couples an audio encoder with a language model and is trained on the NISQA corpus using template-based question-answer pairs covering overall MOS and four perceptual dimensions (noisiness, coloration, discontinuity, and loudness) in both single-ended (degraded only) and double-ended (degraded plus clean reference) setups. Instead of directly regressing scores, SpeechQualityLLM is supervised to generate textual answers from which numeric predictions are parsed and evaluated with standard regression and ranking metrics; on held-out NISQA clips, the double-ended model attains a MOS mean absolute error (MAE) of approximately 0.41 with Pearson correlation of 0.86, with competitive performance on dimension-wise tasks. Beyond these quantitative gains, SpeechQualityLLM offers a flexible natural-language interface in which the language model acts as an audio quality expert: practitioners can query arbitrary aspects of degradations, prompt the model to emulate different listener profiles to capture human variability and produce diverse but plausible judgments rather than a single deterministic score, and thereby reduce reliance on large-scale crowdsourced tests and their monetary cost. W e provide a general pipeline for adapting large language models to specialized audio quality assessment tasks via lightweight mul-timodal alignment. Code, model weights, and experimental results are available at GitHub.