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1. Bridging Continuous and Discrete Spaces: Interpretable Sentence Representation Learning via Compositional Operations

   https://doi.org/10.18653/v1/2023.emnlp-main.900  

   Huang, James; Yao, Wenlin; Song, Kaiqiang; Zhang, Hongming; Chen, Muhao; Yu, Dong (January 2023, Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing)

   Traditional sentence embedding models encode sentences into vector representations to capture useful properties such as the semantic similarity between sentences. However, in addition to similarity, sentence semantics can also be interpreted via compositional operations such as sentence fusion or difference. It is unclear whether the compositional semantics of sentences can be directly reflected as compositional operations in the embedding space. To more effectively bridge the continuous embedding and discrete text spaces, we explore the plausibility of incorporating various compositional properties into the sentence embedding space that allows us to interpret embedding transformations as compositional sentence operations. We propose InterSent, an end-to-end framework for learning interpretable sentence embeddings that supports compositional sentence operations in the embedding space. Our method optimizes operator networks and a bottleneck encoder-decoder model to produce meaningful and interpretable sentence embeddings. Experimental results demonstrate that our method significantly improves the interpretability of sentence embeddings on four textual generation tasks over existing approaches while maintaining strong performance on traditional semantic similarity tasks. 

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2. Fused Gromov-Wasserstein Variance Decomposition with Linear Optimal Transport

   Wilson, Michael; Needham, Tom; Srivastava, Anuj (November 2024, arXiv.org)

   Wasserstein distances form a family of metrics on spaces of probability measures that have recently seen many applications. However, statistical analysis in these spaces is complex due to the nonlinearity of Wasserstein spaces. One potential solution to this problem is Linear Optimal Transport (LOT). This method allows one to find a Euclidean embedding, called {\it LOT embedding}, of measures in some Wasserstein spaces, but some information is lost in this embedding. So, to understand whether statistical analysis relying on LOT embeddings can make valid inferences about original data, it is helpful to quantify how well these embeddings describe that data. To answer this question, we present a decomposition of the {\it Fr\'echet variance} of a set of measures in the 2-Wasserstein space, which allows one to compute the percentage of variance explained by LOT embeddings of those measures. We then extend this decomposition to the Fused Gromov-Wasserstein setting. We also present several experiments that explore the relationship between the dimension of the LOT embedding, the percentage of variance explained by the embedding, and the classification accuracy of machine learning classifiers built on the embedded data. We use the MNIST handwritten digits dataset, IMDB-50000 dataset, and Diffusion Tensor MRI images for these experiments. Our results illustrate the effectiveness of low dimensional LOT embeddings in terms of the percentage of variance explained and the classification accuracy of models built on the embedded data. 

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3. Demystifying Cipher-Following in Large Language Models via Activation Analysis

   Gross, Megan; Kaya, Yigitcan; Kruegel, Chrisopher; Vigna, Giovanni (December 2025, Mechanistic Interpretability Workshop at NeurIPS 2025)

   Cipher transformations have been studied historically in cryptography, but little work has explored how large language models (LLMs) represent and process them. We evaluate the ability of three models: Llama 3.1, Gemma 2, and Qwen 3 on performing translation and dictionary tasks across ten cipher systems from a variety of families, and compare it against a commercially available model, GPT-5. Beyond task performance, we analyze embedding spaces of Llama variants to explore whether ciphers are internalized similarly to languages. Our findings suggest that cipher embeddings cluster together and, in some cases, overlap with lowerresource or less frequently represented languages. Steering-vector experiments further reveal that adjusting cipher-related directions in latent space can shift outputs toward these languages, suggesting shared representational structures. This study provides an initial framework for understanding how LLMs encode ciphers, bridging interpretability, and security. By framing ciphers in a similar way to languages, we highlight new directions for model analysis and for designing defenses against cipher-based jailbreaking attacks. We share our source code and data for reproduction research on GitHub. 

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4. On Harmonic Hilbert Spaces on Compact Abelian Groups

   https://doi.org/10.1007/s00041-023-09992-4  

   Das, Suddhasattwa; Giannakis, Dimitrios (February 2023, Journal of Fourier Analysis and Applications)

   Abstract Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted$$L^2$$ $L^2$spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach$$^*$$ ${}^{\ast }$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian groupGto have the same spectrum as the$$C^*$$ $C^{\ast }$-algebra of continuous functions onG. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on$$L^2(G)$$ $L^2\left(G\right)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation. 

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5. Learning Hyperbolic Embedding for Phylogenetic Tree Placement and Updates

   https://doi.org/10.3390/biology11091256  

   Jiang, Yueyu; Tabaghi, Puoya; Mirarab, Siavash (September 2022, Biology)

   Phylogenetic placement, used widely in ecological analyses, seeks to add a new species to an existing tree. A deep learning approach was previously proposed to estimate the distance between query and backbone species by building a map from gene sequences to a high-dimensional space that preserves species tree distances. They then use a distance-based placement method to place the queries on that species tree. In this paper, we examine the appropriate geometry for faithfully representing tree distances while embedding gene sequences. Theory predicts that hyperbolic spaces should provide a drastic reduction in distance distortion compared to the conventional Euclidean space. Nevertheless, hyperbolic embedding imposes its own unique challenges related to arithmetic operations, exponentially-growing functions, and limited bit precision, and we address these challenges. Our results confirm that hyperbolic embeddings have substantially lower distance errors than Euclidean space. However, these better-estimated distances do not always lead to better phylogenetic placement. We then show that the deep learning framework can be used not just to place on a backbone tree but to update it to obtain a fully resolved tree. With our hyperbolic embedding framework, species trees can be updated remarkably accurately with only a handful of genes. 

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