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Dimensionality Reduction is one of the charting options available in the the Charts Panel. It displays the relationships between data points in a reduced dimensional space, typically plotting the first two principal components or manifold embeddings on the x- and y-axes, with each dot representing an individual data point or object from the dataset.

To display the Dimensionality Reduction plot, open the Charts Panel and select the Dimensionality Reduction plot option option Image Modified from the Charts selection dropdown menu.

For detailed information about dimensionality reduction in generalgeneral, please refer to the Wikipedia entry for for Dimensionality Reduction.

Appearance

When the Dimensionality Reduction panel is opened while an image with objects and measurements is open, additional parameters must be supplied depending on the method of dimensionality reduction desired. Available options are UMAP¹, PaCMAP², and t-SNE³.  Once the dataset has been reduced in dimensionality, the resulting 2D chart will appear. A legend on the right of the chart will show all object sets/groups and their colors.

General usage

Select Measurements

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Additional methods can be selected under the Advanced drop down arrow. Available methods include UMAP, PaCMAP, and t-SNE. Additional information on parameter tuning for each of the individual methods can be found on their respective wiki pages.

Un-/ Select object sets/groups

To remove an object set/group from being displayed in the chart, unselect it from the legend. To show an object set/group, reselect it.

Select objects

Objects on the plot can be selected by dragging across the chart to create a box or by a polygon: draw individual points within the chart - subsequent points will be connected by a line, when clicking near the first point, the polygon will be created and enclosed points will be selected. You can also select objects by clicking on them individually and Shift + click to select multiple objects. Selected objects appear with red or green circle around the edge. Hovering over an individual data point will bring up the object name as well as the chart data for that point.

Panning

For panning, click and hold right mouse button anywhere within the chart area and move the mouse.

Reduce displayed points

If the number of points in the scatter plot exceeds 10000, the number of points being displayed will automatically be reduced to 10000 and the Reduce data points-button  will appear in the tool bar. Pushing this button evokes a slider which can be used to reduce/increase the number of points shown in the plot. Points are reduced by density ordered quad tree - favoring removal of points in high-density areas. At the bottom of the chart the faction of points currently shown will be indicated. Selecting points by box or polygon will select all points within the area - whether they are currently shown or removed by this point-reduction method.

Context menu options

You can bring up the context menu by right-clicking anywhere in the Charts Panel. The context menu is specific to the Scatter plot and can be used to adjust chart display options.

Methods for Dimensionality Reduction

UMAP (Uniform Manifold Approximation and Projection)

UMAP (Uniform Manifold Approximation and Projection)¹ is a modern dimensionality reduction technique primarily use for visualization of high-dimensional datasets. It is similar in purpose to t-SNE, but is generally faster and can handle larger datasets. UMAP operates on the principle of preserving the topological structure of the data through leveraging Riemannian geometry and algebraic topology to approximate the underlying manifold of the data. By capturing both local and global structures, it provides a comprehensive view of data clusters and relationships.

Parameters

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Advantages

Preserves local and global structure

UMAP can capture non-linear relationships in data, making it effective for complex datasets. It retains both local and global structures in the data, making it suitable for a wide range of applications.

Speed and scalability

UMAP is computationally efficient and scales well to large datasets compared to some other dimensionality reduction methods like t-SNE⁴.

Parameter tuning

UMAP offers flexibility in parameter tuning, allowing users to fine-tune the trade-off between preserving local and global structures.

Disadvantages

Interpretability

The UMAP embedding may not be as interpretable as some other methods, like PCA, which provide more intuitive insights into the data.

Sensitivity to hyperparameters

UMAP's performance can be sensitive to hyperparameter choices, and finding the right parameters may require experimentation.

Limited in high dimensions

Like many dimensionality reduction methods, UMAP can suffer from the curse of dimensionality and may not work well in very high-dimensional spaces.

Computational resource requirements

While UMAP is generally faster than t-SNE, it can still be computationally expensive for extremely large datasets.

PaCMAP (Pairwise Controlled Manifold Approximation)

PaCMAP (Pairwise Controlled Manifold Approximation)² is a dimensionality reduction technique introduced as an alternative to methods like t-SNE and UMAP. The method is designed to balance between preserving local and global structures in the data, addressing some of the other challenges observed with other techniques. It introduces a pairwise attraction and repulsion term to control the balance during manifold learning, and is noted for its speed and ability to handle large datasets while producing interpretable embeddings.

Parameters

Advantages

Hybrid Approach

PacMAP incorporates both local and global structure preservation, aiming to capture the best of both worlds from methods like t-SNE (local) and PCA (global). PacMAP is designed to combine the advantages of t-SNE (local structure preservation) and UMAP/PCA (global structure preservation).

Flexibility in preserving local and global structure

The method can be adjusted to emphasize either local or global structures, depending on the nature of the data and the user's goals.

Reduced Crowding Problem

The method seeks to alleviate the "crowding problem" common in t-SNE, which can cause clusters to be pushed too far apart.

Reduced Stochasticity

Offers more consistent results across runs compared to the stochastic nature of t-SNE. While there are parameters to tweak, the method is designed to be more robust to parameter changes than t-SNE.

Disadvantages

Complexity and familiarity

Being a hybrid method, PaCMAP might be harder to understand for users familiar with simpler, single-objective methods. Some data analysis communities might be less familiar with PacMAP, leading to potential challenges in adoption or interpretation. Being newer, it may not have been as extensively vetted in diverse applications as longer-standing methods like t-SNE or PCA.

Parameter Sensitivity

While designed to be more robust to parameter changes, it's still possible for results to vary based on parameter choices. Depending on the data, there is a risk of over-emphasizing either local or global structures if not tuned correctly.

Interpretability

Like other dimensionality reduction techniques, interpreting the reduced dimensions can still be non-intuitive.

t-SNE (t-Stochastic Neighbor Embedding)

t-SNE (t-Stochastic Neighbor Embedding)³ is a popular dimensionality reduction method use for the visualization of high-dimensional data. t-SNE works by preserving the local structure of the data, often resulting in clear separation of clusters. Unlike PCA (Principal Component Analysis) that focuses on maximizing variance, t-SNE emphasizes on keeping similar distances close and dissimilar ones apart in the reduced space. However, due to its emphasis on local structures, it sometimes exaggerates clusters and may not always preserve the global structure of data. This method is computationally intensive, especially for large datasets.

Parameters

Advantages

Preservation of Local Structures

t-SNE excels at preserving local structures in the data, making it effective for identifying clusters of similar data points.

Flexibility

It can handle non-linear data structures effectively, unlike certain linear methods such as PCA.

Visualization

Particularly useful for visualizing high-dimensional data in two or three dimensions.

Disadvantages

Computational Intensity

The algorithm can be computationally expensive, especially with large datasets.

Stochastic Nature

The final visualizations can vary between runs due to the algorithm's randomness, which might lead to inconsistencies.

Hyperparameter Sensitivity

Results can be sensitive to the choice of perplexity.

Interpretability

The distances between clusters in the t-SNE plot do not always have meaningful interpretations. The algorithm prioritizes preserving local structures over global ones. The density of data points in the t-SNE visualization does not necessarily represent the density in the original high-dimensional space.

Optimal for Visualization Only

While great for visualization, t-SNE embeddings may not always be suitable as input for other machine learning algorithms.