Dmitry Kobak

Sebastian Damrich, Jan Niklas Böhm, Fred A. Hamprecht et al.

Neighbor embedding methods $t$-SNE and UMAP are the de facto standard for visualizing high-dimensional datasets. Motivated from entirely different viewpoints, their loss functions appear to be unrelated. In practice, they yield strongly differing embeddings and can suggest conflicting interpretations of the same data. The fundamental reasons for this and, more generally, the exact relationship between $t$-SNE and UMAP have remained unclear. In this work, we uncover their conceptual connection via a new insight into contrastive learning methods. Noise-contrastive estimation can be used to optimize $t$-SNE, while UMAP relies on negative sampling, another contrastive method. We find the precise relationship between these two contrastive methods and provide a mathematical characterization of the distortion introduced by negative sampling. Visually, this distortion results in UMAP generating more compact embeddings with tighter clusters compared to $t$-SNE. We exploit this new conceptual connection to propose and implement a generalization of negative sampling, allowing us to interpolate between (and even extrapolate beyond) $t$-SNE and UMAP and their respective embeddings. Moving along this spectrum of embeddings leads to a trade-off between discrete / local and continuous / global structures, mitigating the risk of over-interpreting ostensible features of any single embedding. We provide a PyTorch implementation.

Jan Niklas Böhm, Philipp Berens, Dmitry Kobak

Visualization methods based on the nearest neighbor graph, such as t-SNE or UMAP, are widely used for visualizing high-dimensional data. Yet, these approaches only produce meaningful results if the nearest neighbors themselves are meaningful. For images represented in pixel space this is not the case, as distances in pixel space are often not capturing our sense of similarity and therefore neighbors are not semantically close. This problem can be circumvented by self-supervised approaches based on contrastive learning, such as SimCLR, relying on data augmentation to generate implicit neighbors, but these methods do not produce two-dimensional embeddings suitable for visualization. Here, we present a new method, called t-SimCNE, for unsupervised visualization of image data. T-SimCNE combines ideas from contrastive learning and neighbor embeddings, and trains a parametric mapping from the high-dimensional pixel space into two dimensions. We show that the resulting 2D embeddings achieve classification accuracy comparable to the state-of-the-art high-dimensional SimCLR representations, thus faithfully capturing semantic relationships. Using t-SimCNE, we obtain informative visualizations of the CIFAR-10 and CIFAR-100 datasets, showing rich cluster structure and highlighting artifacts and outliers.

Fynn Bachmann, Philipp Hennig, Dmitry Kobak

Scientific datasets often have hierarchical structure: for example, in surveys, individual participants (samples) might be grouped at a higher level (units) such as their geographical region. In these settings, the interest is often in exploring the structure on the unit level rather than on the sample level. Units can be compared based on the distance between their means, however this ignores the within-unit distribution of samples. Here we develop an approach for exploratory analysis of hierarchical datasets using the Wasserstein distance metric that takes into account the shapes of within-unit distributions. We use t-SNE to construct 2D embeddings of the units, based on the matrix of pairwise Wasserstein distances between them. The distance matrix can be efficiently computed by approximating each unit with a Gaussian distribution, but we also provide a scalable method to compute exact Wasserstein distances. We use synthetic data to demonstrate the effectiveness of our Wasserstein t-SNE, and apply it to data from the 2017 German parliamentary election, considering polling stations as samples and voting districts as units. The resulting embedding uncovers meaningful structure in the data.

Sebastian Damrich, Philipp Berens, Dmitry Kobak

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.

Rita González-Márquez, Dmitry Kobak

The ICLR conference is unique among the top machine learning conferences in that all submitted papers are openly available. Here we present the ICLR dataset consisting of abstracts of all 24 thousand ICLR submissions from 2017-2024 with meta-data, decision scores, and custom keyword-based labels. We find that on this dataset, bag-of-words representation outperforms most dedicated sentence transformer models in terms of $k$NN classification accuracy, and the top performing language models barely outperform TF-IDF. We see this as a challenge for the NLP community. Furthermore, we use the ICLR dataset to study how the field of machine learning has changed over the last seven years, finding some improvement in gender balance. Using a 2D embedding of the abstracts' texts, we describe a shift in research topics from 2017 to 2024 and identify hedgehogs and foxes among the authors with the highest number of ICLR submissions.

Cyril de Bodt, Alex Diaz-Papkovich, Michael Bleher et al.

Large collections of high-dimensional data have become nearly ubiquitous across many academic fields and application domains, ranging from biology to the humanities. Since working directly with high-dimensional data poses challenges, the demand for algorithms that create low-dimensional representations, or embeddings, for data visualization, exploration, and analysis is now greater than ever. In recent years, numerous embedding algorithms have been developed, and their usage has become widespread in research and industry. This surge of interest has resulted in a large and fragmented research field that faces technical challenges alongside fundamental debates, and it has left practitioners without clear guidance on how to effectively employ existing methods. Aiming to increase coherence and facilitate future work, in this review we provide a detailed and critical overview of recent developments, derive a list of best practices for creating and using low-dimensional embeddings, evaluate popular approaches on a variety of datasets, and discuss the remaining challenges and open problems in the field.

Noël Kury, Dmitry Kobak, Sebastian Damrich

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 seven real-world datasets, including five from single-cell transcriptomics and one from population genetics, showcasing qualitatively and quantitatively its superior ability to preserve structure across multiple scales compared to previous approaches.

Andrew Draganov, Sharvaree Vadgama, Sebastian Damrich et al.

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.

Ifeoma Veronica Nwabufo, Jan Niklas Böhm, Philipp Berens et al.

Self-supervised learning methods based on data augmentations, such as SimCLR, BYOL, or DINO, allow obtaining semantically meaningful representations of image datasets and are widely used prior to supervised fine-tuning. A recent self-supervised learning method, $t$-SimCNE, uses contrastive learning to directly train a 2D representation suitable for visualisation. When applied to natural image datasets, $t$-SimCNE yields 2D visualisations with semantically meaningful clusters. In this work, we used $t$-SimCNE to visualise medical image datasets, including examples from dermatology, histology, and blood microscopy. We found that increasing the set of data augmentations to include arbitrary rotations improved the results in terms of class separability, compared to data augmentations used for natural images. Our 2D representations show medically relevant structures and can be used to aid data exploration and annotation, improving on common approaches for data visualisation.

Rita González-Márquez, Philipp Berens, Dmitry Kobak

Text embeddings, i.e. vector representations of entire texts, play an important role in many NLP applications, such as retrieval-augmented generation, sentiment analysis, clustering, or visualizing collections of texts for data exploration. Currently, top-performing embedding models are derived from pre-trained language models via extensive supervised fine-tuning using curated text pairs. This contrasts with computer vision, where self-supervised training based on data augmentations has demonstrated remarkable success. Here we systematically compare the two most well-known augmentation strategies for positive pair generation in contrastive learning of text embeddings. We assess embedding quality on MTEB and additional in-domain evaluations and show that cropping augmentation strongly outperforms the dropout-based approach. We find that on out-of-domain data, the quality of resulting embeddings is below the supervised SOTA models, but for in-domain data, self-supervised fine-tuning produces high-quality text embeddings after very short fine-tuning, sometimes only marginally below the supervised SOTA. Finally, we show that representation quality increases towards the last transformer layers, which undergo the largest change during fine-tuning; and that fine-tuning only those last layers is sufficient to reach similar embedding quality.

Jan Niklas Böhm, Marius Keute, Alica Guzmán et al.

Graph layouts and node embeddings are two distinct paradigms for non-parametric graph representation learning. In the former, nodes are embedded into 2D space for visualization purposes. In the latter, nodes are embedded into a high-dimensional vector space for downstream processing. State-of-the-art algorithms for these two paradigms, force-directed layouts and random-walk-based contrastive learning (such as DeepWalk and node2vec), have little in common. In this work, we show that both paradigms can be approached with a single coherent framework based on established neighbor embedding methods. Specifically, we introduce graph t-SNE, a neighbor embedding method for two-dimensional graph layouts, and graph CNE, a contrastive neighbor embedding method that produces high-dimensional node representations by optimizing the InfoNCE objective. We show that both graph t-SNE and graph CNE strongly outperform state-of-the-art algorithms in terms of local structure preservation, while being conceptually simpler.

Dmitry Kobak, Rita González-Márquez, Emőke-Ágnes Horvát et al.

Large language models (LLMs) like ChatGPT can generate and revise text with human-level performance. These models come with clear limitations: they can produce inaccurate information, reinforce existing biases, and be easily misused. Yet, many scientists use them for their scholarly writing. But how wide-spread is such LLM usage in the academic literature? To answer this question for the field of biomedical research, we present an unbiased, large-scale approach: we study vocabulary changes in over 15 million biomedical abstracts from 2010--2024 indexed by PubMed, and show how the appearance of LLMs led to an abrupt increase in the frequency of certain style words. This excess word analysis suggests that at least 13.5% of 2024 abstracts were processed with LLMs. This lower bound differed across disciplines, countries, and journals, reaching 40% for some subcorpora. We show that LLMs have had an unprecedented impact on scientific writing in biomedical research, surpassing the effect of major world events such as the Covid pandemic.

Sam Greydanus, Dmitry Kobak

Although deep learning models have taken on commercial and political relevance, key aspects of their training and operation remain poorly understood. This has sparked interest in science of deep learning projects, many of which require large amounts of time, money, and electricity. But how much of this research really needs to occur at scale? In this paper, we introduce MNIST-1D: a minimalist, procedurally generated, low-memory, and low-compute alternative to classic deep learning benchmarks. Although the dimensionality of MNIST-1D is only 40 and its default training set size only 4000, MNIST-1D can be used to study inductive biases of different deep architectures, find lottery tickets, observe deep double descent, metalearn an activation function, and demonstrate guillotine regularization in self-supervised learning. All these experiments can be conducted on a GPU or often even on a CPU within minutes, allowing for fast prototyping, educational use cases, and cutting-edge research on a low budget.

Jan Niklas Böhm, Philipp Berens, Dmitry Kobak

Neighbor embeddings are a family of methods for visualizing complex high-dimensional datasets using $k$NN graphs. To find the low-dimensional embedding, these algorithms combine an attractive force between neighboring pairs of points with a repulsive force between all points. One of the most popular examples of such algorithms is t-SNE. Here we empirically show that changing the balance between the attractive and the repulsive forces in t-SNE using the exaggeration parameter yields a spectrum of embeddings, which is characterized by a simple trade-off: stronger attraction can better represent continuous manifold structures, while stronger repulsion can better represent discrete cluster structures and yields higher $k$NN recall. We find that UMAP embeddings correspond to t-SNE with increased attraction; mathematical analysis shows that this is because the negative sampling optimisation strategy employed by UMAP strongly lowers the effective repulsion. Likewise, ForceAtlas2, commonly used for visualizing developmental single-cell transcriptomic data, yields embeddings corresponding to t-SNE with the attraction increased even more. At the extreme of this spectrum lie Laplacian Eigenmaps. Our results demonstrate that many prominent neighbor embedding algorithms can be placed onto the attraction-repulsion spectrum, and highlight the inherent trade-offs between them.

Yves Bernaerts, Philipp Berens, Dmitry Kobak

Patch-seq, a recently developed experimental technique, allows neuroscientists to obtain transcriptomic and electrophysiological information from the same neurons. Efficiently analyzing and visualizing such paired multivariate data in order to extract biologically meaningful interpretations has, however, remained a challenge. Here, we use sparse deep neural networks with and without a two-dimensional bottleneck to predict electrophysiological features from the transcriptomic ones using a group lasso penalty, yielding concise and biologically interpretable two-dimensional visualizations. In two large example data sets, this visualization reveals known neural classes and their marker genes without biological prior knowledge. We also demonstrate that our method is applicable to other kinds of multimodal data, such as paired transcriptomic and proteomic measurements provided by CITE-seq.

Dmitry Kobak, George Linderman, Stefan Steinerberger et al.

T-distributed stochastic neighbour embedding (t-SNE) is a widely used data visualisation technique. It differs from its predecessor SNE by the low-dimensional similarity kernel: the Gaussian kernel was replaced by the heavy-tailed Cauchy kernel, solving the "crowding problem" of SNE. Here, we develop an efficient implementation of t-SNE for a $t$-distribution kernel with an arbitrary degree of freedom $ν$, with $ν\to\infty$ corresponding to SNE and $ν=1$ corresponding to the standard t-SNE. Using theoretical analysis and toy examples, we show that $ν<1$ can further reduce the crowding problem and reveal finer cluster structure that is invisible in standard t-SNE. We further demonstrate the striking effect of heavier-tailed kernels on large real-life data sets such as MNIST, single-cell RNA-sequencing data, and the HathiTrust library. We use domain knowledge to confirm that the revealed clusters are meaningful. Overall, we argue that modifying the tail heaviness of the t-SNE kernel can yield additional insight into the cluster structure of the data.

Dmitry Kobak, Jonathan Lomond, Benoit Sanchez

A conventional wisdom in statistical learning is that large models require strong regularization to prevent overfitting. Here we show that this rule can be violated by linear regression in the underdetermined $n\ll p$ situation under realistic conditions. Using simulations and real-life high-dimensional data sets, we demonstrate that an explicit positive ridge penalty can fail to provide any improvement over the minimum-norm least squares estimator. Moreover, the optimal value of ridge penalty in this situation can be negative. This happens when the high-variance directions in the predictor space can predict the response variable, which is often the case in the real-world high-dimensional data. In this regime, low-variance directions provide an implicit ridge regularization and can make any further positive ridge penalty detrimental. We prove that augmenting any linear model with random covariates and using minimum-norm estimator is asymptotically equivalent to adding the ridge penalty. We use a spiked covariance model as an analytically tractable example and prove that the optimal ridge penalty in this case is negative when $n\ll p$.

Dmitry Kobak, Wieland Brendel, Christos Constantinidis et al.

Neurons in higher cortical areas, such as the prefrontal cortex, are known to be tuned to a variety of sensory and motor variables. The resulting diversity of neural tuning often obscures the represented information. Here we introduce a novel dimensionality reduction technique, demixed principal component analysis (dPCA), which automatically discovers and highlights the essential features in complex population activities. We reanalyze population data from the prefrontal areas of rats and monkeys performing a variety of working memory and decision-making tasks. In each case, dPCA summarizes the relevant features of the population response in a single figure. The population activity is decomposed into a few demixed components that capture most of the variance in the data and that highlight dynamic tuning of the population to various task parameters, such as stimuli, decisions, rewards, etc. Moreover, dPCA reveals strong, condition-independent components of the population activity that remain unnoticed with conventional approaches.