This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues.

Here, we include implementation details omitted from the main paper for brevity. Upon acceptance, a deanonymized repository will be released. The last layer's dimension depended upon the exact The feature extractors and decoders varied by domain. In particular, we found that if we did not apply this linear transformation (i.e., pass the raw encodings For VQ-based methods, use a large enough codebook to have at least one element per class. Other differences simply reflected differences in architecture (e.g., For iNat, we trained all models with batch size 256, using the hyperparameters specified in Table 3.

By composing graphical models with deep learning architectures, we learn generative models with the strengths of both frameworks. The structured variational autoencoder (SVAE) inherits structure and interpretability from graphical models, and flexible likelihoods for high-dimensional data from deep learning, but poses substantial optimization challenges. We propose novel algorithms for learning SVAEs, and are the first to demonstrate the SVAE's ability to handle multimodal uncertainty when data is missing by incorporating discrete latent variables. Our memory-efficient implicit differentiation scheme makes the SVAE tractable to learn via gradient descent, while demonstrating robustness to incomplete optimization. To more rapidly learn accurate graphical model parameters, we derive a method for computing natural gradients without manual derivations, which avoids biases found in prior work. These optimization innovations enable the first comparisons of the SVAE to state-of-the-art time series models, where the SVAE performs competitively while learning interpretable and structured discrete data representations.

Neural networks often learn task-specific latent representations that fail to generalize to novel settings or tasks. Conversely, humans learn discrete representations (i.e., concepts or words) at a variety of abstraction levels (e.g., bird vs. sparrow') and use the appropriate abstraction based on tasks. Inspired by this, we train neural models to generate a spectrum of discrete representations, and control the complexity of the representations (roughly, how many bits are allocated for encoding inputs) by tuning the entropy of the distribution over representations. In finetuning experiments, using only a small number of labeled examples for a new task, we show that (1) tuning the representation to a task-appropriate complexity level supports the greatest finetuning performance, and (2) in a human-participant study, users were able to identify the appropriate complexity level for a downstream task via visualizations of discrete representations. Our results indicate a promising direction for rapid model finetuning by leveraging human insight.

Offline Reinforcement Learning (RL) has demonstrated promising results in various applications by learning policies from previously collected datasets, reducing the need for online exploration and interactions. However, real-world scenarios usually involve partial observability, which brings crucial challenges of the deployment of offline RL methods: i) the policy trained on data with full observability is not robust against the masked observations during execution, and ii) the information of which parts of observations are masked is usually unknown during training. In order to address these challenges, we present Offline RL with DiscrEte pRoxy representations (ORDER), a probabilistic framework which leverages novel state representations to improve the robustness against diverse masked observabilities. Specifically, we propose a discrete representation of the states and use a proxy representation to recover the states from masked partial observable trajectories. The training of ORDER can be compactly described as the following three steps.

Wastewater-based genomic surveillance has emerged as a powerful tool for population-level viral monitoring, offering comprehensive insights into circulating viral variants across entire communities. However, this approach faces significant computational challenges stemming from high sequencing noise, low viral coverage, fragmented reads, and the complete absence of labeled variant annotations. Traditional reference-based variant calling pipelines struggle with novel mutations and require extensive computational resources. We present a comprehensive framework for unsupervised viral variant detection using Vector-Quantized Variational Autoencoders (VQ-VAE) that learns discrete codebooks of genomic patterns from k-mer tokenized sequences without requiring reference genomes or variant labels. Our approach extends the base VQ-VAE architecture with masked reconstruction pretraining for robustness to missing data and contrastive learning for highly discriminative embeddings. Evaluated on SARS-CoV-2 wastewater sequencing data comprising approximately 100,000 reads, our VQ-VAE achieves 99.52% mean token-level accuracy and 56.33% exact sequence match rate while maintaining 19.73% codebook utilization (101 of 512 codes active), demonstrating efficient discrete representation learning. Contrastive fine-tuning with different projection dimensions yields substantial clustering improvements: 64-dimensional embeddings achieve +35% Silhouette score improvement (0.31 to 0.42), while 128-dimensional embeddings achieve +42% improvement (0.31 to 0.44), clearly demonstrating the impact of embedding dimensionality on variant discrimination capability. Our reference-free framework provides a scalable, interpretable approach to genomic surveillance with direct applications to public health monitoring.

Discrete representation learning has shown promising results across various domains, including generation and understanding in image, speech and language. Inspired by these advances, we propose MuseTok, a tokenization method for symbolic music, and investigate its effectiveness in both music generation and understanding tasks. MuseTok employs the residual vector quantized-variational autoencoder (RQ-VAE) on bar-wise music segments within a Transformer-based encoder-decoder framework, producing music codes that achieve high-fidelity music reconstruction and accurate understanding of music theory. For comprehensive evaluation, we apply MuseTok to music generation and semantic understanding tasks, including melody extraction, chord recognition, and emotion recognition. Models incorporating MuseTok outperform previous representation learning baselines in semantic understanding while maintaining comparable performance in content generation. Furthermore, qualitative analyses on MuseTok codes, using ground-truth categories and synthetic datasets, reveal that MuseTok effectively captures underlying musical concepts from large music collections.