🤖 machine learning

DREAMS: Preserving both Local and Global Structure in Dimensionality Reduction

This paper introduces DREAMS, a dimensionality reduction method that effectively balances local and global structure preservation by combining the strengths of t-SNE and PCA through a simple regularization term, outperforming existing approaches across multiple real-world datasets.

Original authors: Noël Kury, Dmitry Kobak, Sebastian Damrich

Published 2026-06-09

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CC BY 4.0

Original authors: Noël Kury, Dmitry Kobak, Sebastian Damrich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, messy box of thousands of different toys. Some are tiny, intricate action figures; others are large, bulky building blocks. You want to lay them out on a flat table so you can see how they relate to each other.

The problem is that this box is 3D (or even higher dimensions), but your table is only 2D. When you try to flatten it, you have to make a choice:

The "Local" Approach (t-SNE): You decide to keep the tiny action figures grouped tightly together so you can see their individual details. But in doing so, you might accidentally scatter the big building blocks all over the table, losing the sense of the "big picture."
The "Global" Approach (PCA): You decide to keep the big building blocks in their correct, logical positions relative to each other. But in doing so, the tiny action figures get squished together into a single, blurry blob, and you can't tell them apart anymore.

For a long time, scientists had to pick one of these two views. They couldn't have both the fine details and the big picture at the same time.

Enter DREAMS: The "Best of Both Worlds" Method

The paper introduces a new method called DREAMS (Dimensionality Reduction Enhanced Across Multiple Scales). Think of DREAMS as a smart, adjustable lens that lets you see the whole scene clearly while still zooming in on the small details.

Here is how it works, using a simple analogy:

The "Rubber Sheet" and the "Magnet"
Imagine you are stretching a rubber sheet (this is the t-SNE part). You pull the tiny action figures apart so they don't overlap, preserving their local neighborhoods. This is great for seeing the small details.

However, the rubber sheet might stretch too much, making the whole layout look distorted. To fix this, DREAMS adds a magnet underneath the sheet (this is the PCA part). This magnet gently pulls the entire sheet toward a pre-determined, logical shape (the "global structure").

If the magnet is weak: The rubber sheet stretches freely. You get a detailed map of the tiny figures, but the big picture might look a bit wobbly.
If the magnet is strong: The sheet is pulled tight against the logical shape. You get a perfect big picture, but the tiny figures get squished.
The Sweet Spot (The DREAMS Solution): The authors found a "Goldilocks" setting for the magnet's strength. At this setting, the rubber sheet stretches enough to show the tiny figures clearly, but the magnet is strong enough to keep the big building blocks in their correct, logical places.

What Did They Find?

The researchers tested this "adjustable magnet" method on 11 different real-world datasets, including complex biological data (like brain cells) and image data (like handwritten digits).

It Works Better Than the Competition: When they compared DREAMS to other popular methods (like UMAP, t-SNE, and PCA), DREAMS consistently did the best job of keeping both the "neighborhoods" (local structure) and the "continents" (global structure) intact.
A Smooth Spectrum: One of the coolest features is that you can slide a dial (called $\lambda$ ) to change the view. You can slide it all the way to the "Local" side to see fine details, all the way to the "Global" side to see the big picture, or stop anywhere in between to get a perfect balance.
Real-World Example: In a dataset of brain cells, standard methods either showed you the broad categories of cells (neurons vs. non-neurons) but missed the specific sub-types, OR they showed you the specific sub-types but scrambled the broad categories. DREAMS showed you both: the broad groups were in the right places, and the specific cell types were clearly separated within those groups.

The Bottom Line

DREAMS is a new tool that solves an old problem: how to flatten complex, multi-layered data without losing the forest or the trees. By simply adding a "gentle pull" toward a global map while letting the local details breathe, it creates visualizations that are more accurate and easier to understand than previous methods.

The authors note that while it takes a little more computing power to run this "magnet" system, the result is a much clearer picture of complex data, especially when that data has a hierarchy (like a family tree or a city map with neighborhoods and districts).

Technical Summary: DREAMS – Preserving both Local and Global Structure in Dimensionality Reduction

Problem Statement

Dimensionality reduction (DR) is essential for visualizing high-dimensional data, yet existing methods typically excel at preserving either local structure or global structure, but rarely both simultaneously.

Local Structure: Methods like t-SNE and UMAP effectively preserve neighborhood relationships and fine-grained clusters (e.g., specific cell types) but often distort global arrangements (e.g., broad class separations or developmental trajectories).
Global Structure: Methods like PCA and MDS preserve macroscopic data patterns and variance but often fail to resolve fine local relationships, merging distinct clusters.

In real-world datasets, such as single-cell transcriptomics, both scales of structure are meaningful and complementary. Neglecting one scale can lead to incomplete or biased interpretations. Current hybrid approaches often struggle to balance these competing objectives effectively or fail to recover the pure extremes (standard PCA or t-SNE) within their optimization spectra.

Methodology: DREAMS

The authors propose DREAMS (Dimensionality Reduction Enhanced Across Multiple Scales), a non-parametric method that combines the local sensitivity of t-SNE with the global structure preservation of PCA via a simple regularization term.

Core Mechanism

DREAMS augments the standard t-SNE loss function with a quadratic penalty that encourages the low-dimensional embedding to align with a precomputed global reference embedding (default: PCA).

The objective function is defined as:
$L(Y) = (1 - \lambda)L_{t-SNE}(Y) + \lambda \frac{1}{n} \|Y - \alpha \tilde{Y}\|_F^2$

Where:

$Y$ is the low-dimensional embedding being optimized.
$L_{t-SNE}(Y)$ is the standard t-SNE loss (Kullback–Leibler divergence between high-dimensional and low-dimensional similarity distributions).
$\tilde{Y}$ is the precomputed global reference embedding (e.g., PCA).
$\lambda \in [0, 1]$ is a hyperparameter controlling the trade-off between local and global preservation.
$\alpha$ is a scalar scaling factor computed at each gradient descent iteration to match the scale of the t-SNE embedding ( $Y$ ) with the reference embedding ( $\tilde{Y}$ ), ensuring compatibility between the linear PCA scale and the non-linear t-SNE scale.

Implementation Details

Reference Embedding: By default, DREAMS uses PCA due to its speed and interpretability. However, the framework supports other global methods like MDS (referred to as DREAMS-MDS).
Optimization: The method is implemented based on the openTSNE library, adding the regularization term to the gradient-descent optimization.
Alternative Approach: The authors also explored a "Linear Decoding" variant (DREAMS-CNE-Decoder) using a trainable linear decoder to reconstruct the original data from the embedding, but found the precomputed reference approach superior in preserving local structure.
Spectrum: By varying $\lambda$ $λ$ , DREAMS generates a continuous spectrum of embeddings:
- $\lambda = 0$ : Recovers standard t-SNE.
- $\lambda = 1$ : Recovers the scaled PCA embedding.
- $0 < \lambda < 1$ : Provides a balanced trade-off.

Key Contributions

Regularization Strategy: Introduction of a simple, effective regularization term that integrates global structure preservation (via PCA/MDS) directly into the t-SNE objective.
Spectrum of Visualizations: Demonstration of a tunable spectrum of embeddings that offers a state-of-the-art trade-off between local and global structure preservation.
Comprehensive Benchmarking: Evaluation of ten algorithms across eleven real-world datasets, including single-cell transcriptomics, genomic data, and image datasets.
Open-Source Implementation: Provision of an open-source implementation based on openTSNE and CNE (Contrastive Neighbor Embedding) libraries.

Experimental Results

The authors evaluated DREAMS against baselines (t-SNE, UMAP, PCA, MDS) and hybrid methods (TriMap, PaCMAP, PHATE, SQuadMDS-hybrid, StarMAP) using two primary metrics:

KNN (Local): k-nearest neighbor recall.
CPD (Global): Correlation of pairwise distances (Spearman correlation).

Findings

Superior Trade-off: On 10 out of 11 datasets, DREAMS (with default $\lambda=0.15$ ) achieved the highest aggregated local-global score. It consistently maintained KNN scores close to t-SNE while achieving CPD scores close to PCA.
Comparison to Hybrids: DREAMS outperformed other hybrid methods like StarMAP and SQuadMDS-hybrid. Specifically, DREAMS preserved local structure significantly better than StarMAP (which uses UMAP) and offered a more favorable global-local trade-off than SQuadMDS-hybrid.
Spectrum Quality: The spectrum of embeddings generated by DREAMS provided a smoother and more effective transition between local and global extremes compared to competing methods, which often failed to recover the pure PCA or MDS limits or suffered performance degradation at the extremes.
Limitations: On datasets lacking prominent global structure (e.g., MNIST), DREAMS could not simultaneously maximize both metrics, as the global structure itself was weak. In such cases, performance was comparable to other methods.

Significance and Claims

The paper claims that DREAMS addresses a critical gap in dimensionality reduction by enabling the simultaneous preservation of local and global structures without substantial compromise.

Interpretability: By anchoring the t-SNE optimization to a PCA reference, DREAMS produces visualizations that are both locally detailed (revealing fine clusters) and globally coherent (preserving broad class separations), making them more faithful to the underlying data hierarchy.
Flexibility: The method is adaptable; users can tune $\lambda$ to prioritize local or global features based on specific analytical needs, or switch the reference embedding (e.g., to MDS) if PCA is suboptimal for a specific dataset's global geometry.
Practicality: The authors note that while DREAMS introduces a slight computational overhead due to the regularization term, the default setting ( $\lambda=0.15$ ) works robustly across diverse datasets without requiring extensive hyperparameter tuning.

The authors conclude that DREAMS provides a more faithful and interpretable visualization tool, particularly for hierarchical datasets where multiple scales of structure are present.