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Miscellaneous Aspects of Machine Learning · Representation Learning

Tvrtko Tadić, Cassiano O Becker, Jennifer Neville

Random Projections have been widely used to generate embeddings for various graph learning tasks due to their computational efficiency. The majority of applications have been justified through the Johnson-Lindenstrauss Lemma. In this paper, we take a step further and investigate how well dot product and cosine similarity are preserved by random projections when these are applied over the rows of the graph matrix. Our analysis provides new asymptotic and finite-sample results, identifies pathological cases, and tests them with numerical experiments. We specialize our fundamental results to a ranking application by computing the probability of random projections flipping the node ordering induced by their embeddings. We find that, depending on the degree distribution, the method produces especially unreliable embeddings for the dot product, regardless of whether the adjacency or the normalized transition matrix is used. With respect to the statistical noise introduced by random projections, we show that cosine similarity produces remarkably more precise approximations.

Miscellaneous Aspects of Machine Learning · Everything Else

MingYu Lu, Chris Lin, Chanwoo Kim, Su-In Lee

As diffusion models are deployed in real-world settings and their performance driven by training data, appraising the contribution of data contributors is crucial to creating incentives for sharing quality data and to implementing policies for data compensation. Depending on the use case, model performance corresponds to various global properties of the distribution learned by a diffusion model (e.g., overall aesthetic quality). Hence, here we address the problem of attributing global properties of diffusion models to data contributors. The Shapley value provides a principled approach to valuation by uniquely satisfying game-theoretic axioms of fairness. However, estimating Shapley values for diffusion models is computationally impractical because it requires retraining and rerunning inference on many subsets of data contributors. We introduce a method to efficiently retrain and rerun inference for Shapley value estimation, by leveraging model pruning and fine-tuning. We evaluate the utility of our method with three use cases: (i) image quality for a DDPM trained on a CIFAR dataset, (ii) demographic diversity for an LDM trained on CelebA-HQ, and (iii) aesthetic quality for a Stable Diffusion model LoRA-finetuned on Post-Impressionist artworks. Our results empirically demonstrate that our framework can identify important data contributors across global properties, outperforming existing attribution methods for diffusion models.

Miscellaneous Aspects of Machine Learning · Representation Learning

Mohammad Omama, Po-han Li, Sandeep Chinchali

Image retrieval is crucial in robotics and computer vision, with downstream applications in robot place recognition and vision-based product recommendations. Modern retrieval systems face two key challenges: scalability and efficiency.State-of-the-art image retrieval systems train specific neural networks for each dataset, an approach that lacks scalability. Furthermore, since retrieval speed is directly proportional to embedding size, existing systems that use large embeddings lack efficiency. To tackle scalability, recent works propose using off-the-shelf foundation models. However, these models, though applicable across datasets, fall short in achieving performance comparable to that of dataset-specific models. Our key observation is that, while foundation models capture necessary subtleties for effective retrieval, the underlying distribution of their embedding space can negatively impact cosine similarity searches. We introduce Autoencoders with Strong Variance Constraints (AE-SVC), which, when used for projection, significantly improves the performance of foundation models. We provide an in-depth theoretical analysis of AE-SVC. Addressing efficiency, we introduce Single-Shot Similarity Space Distillation ((SS)2D), a novel approach to learn embeddings with adaptive sizes that offers a better trade-off between size and performance. We conducted extensive experiments on four retrieval datasets, including Stan-ford Online Products (SoP) and Pittsburgh30k, using four different off-the-shelf foundation models, including DinoV2 and CLIP. AE-SVC demonstrates up to a 16% improvement in retrieval performance, while (SS)2D shows a further 10% improvement for smaller embedding sizes.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Nazia Tasnim, Bryan Plummer

Incremental learning aims to adapt to new sets of categories over time with minimal computational overhead. Prior work often addresses this task by training efficient task-specific adaptors that modify frozen layer weights or features to capture relevant information without affecting predictions on any previously learned categories. While these adaptors are generally more efficient than finetuning the entire network, they still can require tens to hundreds of thousands task-specific trainable parameters even for relatively small networks. This makes it challenging to operate in resource-constrained environments with high communication costs, such as edge devices or mobile phones. Thus, we propose Reparameterized, Compact weight Adaptation for Sequential Tasks (RECAST), a novel method that dramatically reduces the number of task-specific trainable parameters to fewer than 50 – several orders of magnitude less than competing methods like LoRA. RECAST accomplishes this efficiency by learning to decompose layer weights into a soft parameter-sharing framework consisting of a set of shared weight templates and very few module-specific scaling factors or coefficients. This soft parameter-sharing framework allows for effective task-wise reparameterization by tuning only these coefficients while keeping templates frozen. A key innovation of RECAST is the novel weight reconstruction pipeline called Neural Mimicry, which eliminates the need for pretraining from scratch. This allows for high-fidelity emulation of existing pretrained weights within our framework and provides quick adaptability to any model scale and architecture. Extensive experiments across six diverse datasets demonstrate RECAST outperforms the state-of-the-art by up to ∼ 1.5% and improves baselines by > 3% across various scales, architectures, and parameter spaces. Moreover, we show that RECAST’s architecture-agnostic nature allows for seamless integration with existing methods, further boosting performance.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Yeoreum Lee, Jinwook Jung, Sungyong Baik

Large-scale deep learning models with a pretraining-finetuning paradigm have led to a surge of numerous task-specific models fine-tuned from a common pre-trained model.Recently, several research efforts have been made on merging these large models into a single multi-task model, particularly with simple arithmetic on parameters.Such merging methodology faces a central challenge: interference between model parameters fine-tuned on different tasks.Few recent works have focused on designing a new fine-tuning scheme that can lead to small parameter interference, however at the cost of the performance of each task-specific fine-tuned model and thereby limiting that of a merged model.To improve the performance of a merged model, we note that a fine-tuning scheme should aim for (1) smaller parameter interference and (2) better performance of each fine-tuned model on the corresponding task.In this work, we aim to design a new fine-tuning objective function to work towards these two goals.In the course of this process, we find such objective function to be strikingly similar to sharpness-aware minimization (SAM) objective function, which aims to achieve generalization by finding flat minima.Drawing upon our observation, we propose to fine-tune pre-trained models via sharpness-aware minimization.The experimental and theoretical results showcase the effectiveness and orthogonality of our proposed approach, improving performance upon various merging and fine-tuning methods.Our code is available at https://github.com/baiklab/SAFT-Merge.

Miscellaneous Aspects of Machine Learning · Representation Learning

Chenyu Wang, Sharut Gupta, Xinyi Zhang, Sana Tonekaboni, Stefanie Jegelka, Tommi Jaakkola, Caroline Uhler

Multimodal representation learning seeks to relate and decompose information inherent in multiple modalities. By disentangling modality-specific information from information that is shared across modalities, we can improve interpretability and robustness and enable downstream tasks such as the generation of counterfactual outcomes. Separating the two types of information is challenging since they are often deeply entangled in many real-world applications. We propose $\textbf{Disentangled}$ $\textbf{S}$elf-$\textbf{S}$upervised $\textbf{L}$earning (DisentangledSSL), a novel self-supervised approach for learning disentangled representations. We present a comprehensive analysis of the optimality of each disentangled representation, particularly focusing on the scenario not covered in prior work where the so-called $\textit{Minimum Necessary Information}$ (MNI) point is not attainable. We demonstrate that \algo successfully learns shared and modality-specific features on multiple synthetic and real-world datasets and consistently outperforms baselines on various downstream tasks, including prediction tasks for vision-language data, as well as molecule-phenotype retrieval tasks for biological data.

Miscellaneous Aspects of Machine Learning · Causality

Jong-Hoon Ahn, Akshay Vashist

We address the individualized treatment effect (ITE) estimation problem, focusing on continuous, multidimensional, and time-dependent treatments for precision medicine. The central challenge lies in modeling these complex treatment scenarios while capturing dynamic patient responses and minimizing reliance on control data. We propose the Gaussian Mixture Counterfactual Generator (GMCG), a generative model that transforms the Gaussian mixture model—traditionally a tool for clustering and density estimation—into a new tool explicitly geared toward causal inference. This approach generates robust counterfactuals by effectively handling continuous and multidimensional treatment spaces. We evaluate GMCG on synthetic crossover trial data and simulated datasets, demonstrating its superior performance over existing methods, particularly in scenarios with limited control data. GMCG derives its effectiveness from modeling the joint distribution of covariates, treatments, and outcomes using a latent state vector while employing a conditional distribution of the state vector to suppress confounding and isolate treatment-outcome relationships.

Miscellaneous Aspects of Machine Learning · Sequential, Network, and Time Series Modeling

Yunju Cho, Jay-Yoon Lee

Long-term time series forecasting poses significant challenges due to the complex dynamics and temporal variations, particularly when dealing with unseen patterns and data scarcity. Traditional supervised learning approaches, which rely on cleaned and labeled data, struggle to capture these unseen characteristics, limiting their effectiveness in real-world applications. In this study, we propose a semi-supervised approach that leverages multi-view setting on augmented data without requiring explicit future values as labels to address these limitations. By introducing a consensus promotion framework, our method enhances agreement among multiple single-view models on unseen augmented data. This approach not only improves forecasting accuracy but also mitigates error accumulation in long-horizon predictions. Furthermore, we explore the impact of autoregressive and non-autoregressive decoding schemes on error propagation, demonstrating the robustness of our model in extending prediction horizons. Experimental results show that our proposed method not only surpasses traditional supervised models in accuracy but also exhibits greater robustness when extending the prediction horizon.

Miscellaneous Aspects of Machine Learning · Unsupervised and Semi-supervised Learning

Kira Michaela Düsterwald, Samo Hromadka, Makoto Yamada

The performance of unsupervised methods such as clustering depends on the choice of distance metric between features, or ground metric. Commonly, ground metrics are decided with heuristics or learned via supervised algorithms. However, since many interesting datasets are unlabelled, unsupervised ground metric learning approaches have been introduced. One promising option employs Wasserstein singular vectors (WSVs), which emerge when computing optimal transport distances between features and samples simultaneously. WSVs are effective, but can be prohibitively computationally expensive in some applications: $\mathcal{O}(n^2m^2(n \log(n) + m \log(m))$ for $n$ samples and $m$ features. In this work, we propose to augment the WSV method by embedding samples and features on trees, on which we compute the tree-Wasserstein distance (TWD). We demonstrate theoretically and empirically that the algorithm converges to a better approximation of the standard WSV approach than the best known alternatives, and does so with $\mathcal{O}(n^3+m^3+mn)$ complexity. In addition, we prove that the initial tree structure can be chosen flexibly, since tree geometry does not constrain the richness of the approximation up to the number of edge weights. This proof suggests a fast and recursive algorithm for computing the tree parameter basis set, which we find crucial to realising the efficiency gains at scale. Finally, we employ the tree-WSV algorithm to several single-cell RNA sequencing genomics datasets, demonstrating its scalability and utility for unsupervised cell-type clustering problems. These results poise unsupervised ground metric learning with TWD as a low-rank approximation of WSV with the potential for widespread application.

Miscellaneous Aspects of Machine Learning · Causality

Gideon Stein, Maha Shadaydeh, Jan Blunk, Niklas Penzel, Joachim Denzler

Causal discovery, or identifying causal relationships from observational data, is a notoriously challenging task, with numerous methods proposed to tackle it.Despite this, in-the-wild evaluation of these methods is still lacking, as works frequently rely on synthetic data evaluation and sparse real-world examples under critical theoretical assumptions.Real-world causal structures, however, are often complex, evolving over time, non-linear, and influenced by unobserved factors, makingit hard to decide on a proper causal discovery strategy.To bridge this gap, we introduce CausalRivers, the largest in-the-wild causal discovery benchmarking kit for time-series data to date.CausalRivers features an extensive dataset on river discharge that covers the eastern German territory (666 measurement stations) and the state of Bavaria (494 measurement stations).It spans the years 2019 to 2023 with a 15-minute temporal resolution.Further, we provide additional data from a flood around the Elbe River, as an event with a pronounced distributional shift.Leveraging multiple sources of information and time-series meta-data, we constructed two distinct causal ground truth graphs (Bavaria and eastern Germany).These graphs can be sampled to generate thousands of subgraphs to benchmark causal discovery across diverse and challenging settings.To demonstrate the utility of CausalRivers, we evaluate several causal discovery approaches through a set of experiments to identify areas for improvement.CausalRivers has the potential to facilitate robust evaluations and comparisons of causal discovery methods.Besides this primary purpose, we also expect that this dataset will be relevant for connected areas of research, such as time-series forecasting and anomaly detection.Based on this, we hope to push benchmark-driven method development that fosters advanced techniques for causal discovery, as is the case for many other areas of machine learning.

Miscellaneous Aspects of Machine Learning · Scalable Algorithms

Xinran Liu, Rocio Diaz Martin, Yikun Bai, Ashkan Shahbazi, Matthew Thorpe, Akram Aldroubi, Soheil Kolouri

The optimal transport (OT) problem has gained significant traction in modern machine learning for its ability to: (1) provide versatile metrics, such as Wasserstein distances and their variants, and (2) determine optimal couplings between probability measures. To reduce the computational complexity of OT solvers, methods like entropic regularization and sliced optimal transport have been proposed. The sliced OT framework improves efficiency by comparing one-dimensional projections (slices) of high-dimensional distributions. However, despite their computational efficiency, sliced-Wasserstein approaches lack a transportation plan between the input measures, limiting their use in scenarios requiring explicit coupling. In this paper, we address two key questions: Can a transportation plan be constructed between two probability measures using the sliced transport framework? If so, can this plan be used to define a metric between the measures? We propose a ‘lifting’ operation to extend one-dimensional optimal transport plans back to the original space of the measures. By computing the expectation of these lifted plans, we derive a new transportation plan, termed expected sliced transport (EST) plans. We further prove that using the EST plan to weight the sum of the individual Euclidean costs $\|x - y\|^p$ for moving from $x$ to $y$ results in a valid metric between the input discrete probability measures. Finally, we demonstrate the connection between our approach and the recently proposed min-SWGG, along with illustrative numerical examples that support our theoretical findings.

Miscellaneous Aspects of Machine Learning · General Machine Learning Techniques

Guangkun Nie, Gongzheng Tang, Shenda Hong

Imbalanced data distributions are prevalent in real-world scenarios, presenting significant challenges in both classification and regression tasks. This imbalance often causes deep learning models to overfit in regions with abundant data (manyshot regions) while underperforming in regions with sparse data (few-shot regions). Such characteristics limit the applicability of deep learning models across various domains, notably in healthcare, where rare cases often carry greater clinical significance. While recent studies have highlighted the benefits of incorporating distributional information in imbalanced classification tasks, similar strategies have been largely unexplored in imbalanced regression. To address this gap, we propose Dist Loss, a novel loss function that integrates distributional information into model training by jointly optimizing the distribution distance between model predictions and target labels, alongside sample-wise prediction errors. This dual-objective approach encourages the model to balance its predictions across different label regions, leading to significant improvements in accuracy in fewshot regions. We conduct extensive experiments across three datasets spanning computer vision and healthcare: IMDB-WIKI-DIR, AgeDB-DIR, and ECG-KDIR. The results demonstrate that Dist Loss effectively mitigates the impact of imbalanced data distributions, achieving state-of-the-art performance in few-shot regions. Furthermore, Dist Loss is easy to integrate and complements existing methods. To facilitate further research, we provide our implementation at https://github.com/Ngk03/DIR-Dist-Loss.

Miscellaneous Aspects of Machine Learning · Online Learning, Active Learning and Bandits

Evgenii Dzhivelikian, Petr Kuderov, Aleksandr Panov

This paper presents a novel approach to address the challenge of online sequence learning for decision making under uncertainty in non-stationary, partially observable environments. The proposed algorithm, Distributed Hebbian Temporal Memory (DHTM), is based on the factor graph formalism and a multi-component neuron model. DHTM aims to capture sequential data relationships and make cumulative predictions about future observations, forming Successor Features (SFs). Inspired by neurophysiological models of the neocortex, the algorithm uses distributed representations, sparse transition matrices, and local Hebbian-like learning rules to overcome the instability and slow learning of traditional temporal memory algorithms such as RNN and HMM. Experimental results show that DHTM outperforms LSTM, RWKV and a biologically inspired HMM-like algorithm, CSCG, on non-stationary data sets. Our results suggest that DHTM is a promising approach to address the challenges of online sequence learning and planning in dynamic environments.

Miscellaneous Aspects of Machine Learning · Kernel methods

SakethaNath Jagarlapudi, Pratik Jawanpuria, Piyushi Manupriya

We study the unbalanced optimal transport (UOT) problem, where the marginal constraints are enforced using Maximum Mean Discrepancy (MMD) regularization. Our work is motivated by the observation that the literature on UOT is focused on regularization based on $\phi$-divergence (e.g., KL divergence). Despite the popularity of MMD, its role as a regularizer in the context of UOT seems less understood. We begin by deriving a specific dual of MMD-regularized UOT (MMD-UOT), which helps us prove several useful properties. One interesting outcome of this duality result is that MMD-UOT induces novel metrics, which not only lift the ground metric like the Wasserstein but are also sample-wise efficient to estimate like the MMD. Further, for real-world applications involving non-discrete measures, we present an estimator for the transport plan that is supported only on the given ($m$) samples. Under certain conditions, we prove that the estimation error with this finitely-supported transport plan is also $\mathcal{O}(1/\sqrt{m})$. As far as we know, such error bounds that are free from the curse of dimensionality are not known for $\phi$-divergence regularized UOT. Finally, we discuss how the proposed estimator can be computed efficiently using accelerated gradient descent. Our experiments show that MMD-UOT consistently outperforms popular baselines, including KL-regularized UOT and MMD, in diverse machine learning applications.

Miscellaneous Aspects of Machine Learning · Causality

Arvind Raghavan, Elias Bareinboim

It is commonly believed that, in a real-world environment, samples can only be drawn from observational and interventional distributions, corresponding to Layers 1 and 2 of the *Pearl Causal Hierarchy*. Layer 3, representing counterfactual distributions, is believed to be inaccessible by definition. However, Bareinboim, Forney, and Pearl (2015) introduced a procedure that allows an agent to sample directly from a counterfactual distribution, leaving open the question of what other counterfactual quantities can be estimated directly via physical experimentation. We resolve this by introducing a formal definition of realizability, the ability to draw samples from a distribution, and then developing a complete algorithm to determine whether an arbitrary counterfactual distribution is realizable given fundamental physical constraints, such as the inability to go back in time and subject the same unit to a different experimental condition. We illustrate the implications of this new framework for counterfactual data collection using motivating examples from causal fairness and causal reinforcement learning. While the baseline approach in these motivating settings typically follows an interventional or observational strategy, we show that a counterfactual strategy provably dominates both.

Miscellaneous Aspects of Machine Learning · General Machine Learning Techniques

Harshit Kumar, Beomseok Kang, Biswadeep Chakraborty, Saibal Mukhopadhyay

This paper presents the first systematic study of evaluating Deep Neural Networks (DNNs) designed to forecast the evolution of stochastic complex systems. We show that traditional evaluation methods like threshold-based classification metrics and error-based scoring rules assess a DNN's ability to replicate the observed ground truth but fail to measure the DNN's learning of the underlying stochastic process. To address this gap, we propose a new evaluation criteria called _Fidelity to Stochastic Process (F2SP)_, representing the DNN's ability to predict the system property _Statistic-GT_—the ground truth of the stochastic process—and introduce an evaluation metric that exclusively assesses F2SP. We formalize F2SP within a stochastic framework and establish criteria for validly measuring it. We formally show that Expected Calibration Error (ECE) satisfies the necessary condition for testing F2SP, unlike traditional evaluation methods. Empirical experiments on synthetic datasets, including wildfire, host-pathogen, and stock market models, demonstrate that ECE uniquely captures F2SP. We further extend our study to real-world wildfire data, highlighting the limitations of conventional evaluation and discuss the practical utility of incorporating F2SP into model assessment. This work offers a new perspective on evaluating DNNs modeling complex systems by emphasizing the importance of capturing underlying the stochastic process.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Sagi Shaier, Francisco Pereira, Katharina Kann, Lawrence E Hunter, Matt Jones

The evolution of biological neural systems has led to both modularity and sparse coding, which enables energy efficiency and robustness across the diversity of tasks in the lifespan. In contrast, standard neural networks rely on dense, non-specialized architectures, where all model parameters are simultaneously updated to learn multiple tasks, leading to interference. Current sparse neural network approaches aim to alleviate this issue but are hindered by limitations such as 1) trainable gating functions that cause representation collapse, 2) disjoint experts that result in redundant computation and slow learning, and 3) reliance on explicit input or task IDs that limit flexibility and scalability.In this paper we propose Conditionally Overlapping Mixture of ExperTs (COMET), a general deep learning method that addresses these challenges by inducing a modular, sparse architecture with an exponential number of overlapping experts. COMET replaces the trainable gating function used in Sparse Mixture of Experts with a fixed, biologically inspired random projection applied to individual input representations. This design causes the degree of expert overlap to depend on input similarity, so that similar inputs tend to share more parameters. This results in faster learning per update step and improved out-of-sample generalization. We demonstrate the effectiveness of COMET on a range of tasks, including image classification, language modeling, and regression, using several popular deep learning architectures.

Miscellaneous Aspects of Machine Learning · Representation Learning

Manuel Faysse, Hugues Sibille, Tony Wu, Bilel Omrani, Gautier Viaud, CELINE HUDELOT, Pierre Colombo

Documents are visually rich structures that convey information through text, but also figures, page layouts, tables, or even fonts. Since modern retrieval systems mainly rely on the textual information they extract from document pages to index documents -often through lengthy and brittle processes-, they struggle to exploit key visual cues efficiently. This limits their capabilities in many practical document retrieval applications such as Retrieval Augmented Generation (RAG).To benchmark current systems on visually rich document retrieval, we introduce the Visual Document Retrieval Benchmark $\textit{ViDoRe}$, composed of various page-level retrieval tasks spanning multiple domains, languages, and practical settings. The inherent complexity and performance shortcomings of modern systems motivate a new concept; doing document retrieval by directly embedding the images of the document pages. We release $\textit{ColPali}$, a Vision Language Model trained to produce high-quality multi-vector embeddings from images of document pages. Combined with a late interaction matching mechanism, $\textit{ColPali}$ largely outperforms modern document retrieval pipelines while being drastically simpler, faster and end-to-end trainable. We release models, data, code and benchmarks under open licenses at https://hf.co/vidore.

Miscellaneous Aspects of Machine Learning · Representation Learning

Yiqun Sun, Qiang Huang, Yixuan Tang, Anthony Tung, Jun Yu

Semantic text embedding is essential to many tasks in Natural Language Processing (NLP). While black-box models are capable of generating high-quality embeddings, their lack of interpretability limits their use in tasks that demand transparency. Recent approaches have improved interpretability by leveraging domain-expert-crafted or LLM-generated questions, but these methods rely heavily on expert input or well-prompt design, which restricts their generalizability and ability to generate discriminative questions across a wide range of tasks. To address these challenges, we introduce \algo{CQG-MBQA} (Contrastive Question Generation - Multi-task Binary Question Answering), a general framework for producing interpretable semantic text embeddings across diverse tasks. Our framework systematically generates highly discriminative, low cognitive load yes/no questions through the \algo{CQG} method and answers them efficiently with the \algo{MBQA} model, resulting in interpretable embeddings in a cost-effective manner. We validate the effectiveness and interpretability of \algo{CQG-MBQA} through extensive experiments and ablation studies, demonstrating that it delivers embedding quality comparable to many advanced black-box models while maintaining inherently interpretability. Additionally, \algo{CQG-MBQA} outperforms other interpretable text embedding methods across various downstream tasks. The source code is available at \url{https://github.com/dukesun99/CQG-MBQA}.

Miscellaneous Aspects of Machine Learning · Representation Learning

Lorenzo Basile, Santiago Acevedo, Luca Bortolussi, Fabio Anselmi, Alex Rodriguez

To gain insight into the mechanisms behind machine learning methods, it is crucial to establish connections among the features describing data points. However, these correlations often exhibit a high-dimensional and strongly nonlinear nature, which makes them challenging to detect using standard methods. This paper exploits the entanglement between intrinsic dimensionality and correlation to propose a metric that quantifies the (potentially nonlinear) correlation between high-dimensional manifolds. We first validate our method on synthetic data in controlled environments, showcasing its advantages and drawbacks compared to existing techniques. Subsequently, we extend our analysis to large-scale applications in neural network representations. Specifically, we focus on latent representations of multimodal data, uncovering clear correlations between paired visual and textual embeddings, whereas existing methods struggle significantly in detecting similarity. Our results indicate the presence of highly nonlinear correlation patterns between latent manifolds.