Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding (1712.09005v1)

Abstract: t-distributed Stochastic Neighborhood Embedding (t-SNE) is a method for dimensionality reduction and visualization that has become widely popular in recent years. Efficient implementations of t-SNE are available, but they scale poorly to datasets with hundreds of thousands to millions of high dimensional data-points. We present Fast Fourier Transform-accelerated Interpolation-based t-SNE (FIt-SNE), which dramatically accelerates the computation of t-SNE. The most time-consuming step of t-SNE is a convolution that we accelerate by interpolating onto an equispaced grid and subsequently using the fast Fourier transform to perform the convolution. We also optimize the computation of input similarities in high dimensions using multi-threaded approximate nearest neighbors. We further present a modification to t-SNE called "late exaggeration," which allows for easier identification of clusters in t-SNE embeddings. Finally, for datasets that cannot be loaded into the memory, we present out-of-core randomized principal component analysis (oocPCA), so that the top principal components of a dataset can be computed without ever fully loading the matrix, hence allowing for t-SNE of large datasets to be computed on resource-limited machines.

• The paper introduces FIt-SNE, which integrates FFT for acceleration and achieves up to 30x speedup on large datasets.
• FIt-SNE employs multi-threaded approximate nearest neighbor methods and out-of-core PCA to optimize similarity computations and manage memory constraints.
• The approach incorporates a late exaggeration technique that enhances cluster separation during embedding, leading to improved visualization quality.

Efficient Algorithms for t-Distributed Stochastic Neighborhood Embedding (t-SNE)

The paper presents an advanced methodology for accelerating the popular dimensionality reduction technique known as t-distributed Stochastic Neighborhood Embedding (t-SNE). t-SNE is widely employed for visualizing large high-dimensional datasets. Despite its utility, traditional implementations of t-SNE encounter significant computational inefficiency when scaled to datasets ranging into hundreds of thousands or millions of data points. The authors introduce an efficient approach named Fast Fourier Transform-accelerated Interpolation-based t-SNE (FIt-SNE) to address these scaling issues.

Key Innovations

• FFT-accelerated Interpolation: The research prominently features a novel use of Fast Fourier Transforms (FFT) to efficiently compute the convolution integral pivotal in t-SNE's core algorithmic steps. This optimization accelerates the computation of the repulsive forces in the gradient descent process inherent to t-SNE.
• Efficient Computation of Input Similarities: FIt-SNE integrates multi-threaded approximate nearest neighbor methods to optimize the calculation of input similarities in high dimensions, which bracket the complexity of attractive forces in the gradient descent employed by t-SNE. This approach leans on recent insights that suggest fewer neighbors can effectively capture the manifold structure, thus reducing computational overhead significantly.
• Out-of-core PCA: To handle datasets that exceed memory constraints typically faced in high-dimensional analysis, the authors propose an out-of-core implementation for Principal Component Analysis (PCA). This method allows processing data that cannot be fully loaded into the primary memory, thus facilitating t-SNE's application to massive datasets using standard computational resources.
• Late Exaggeration Modification: An additional modification, termed "late exaggeration," is introduced. This method alters the exaggeration of attractive forces towards the last iterations of the embedding process, enhancing the separation of clusters within the embedding.

Numerical Results and Practical Implications

The paper reports strong numerical results demonstrating that FIt-SNE significantly speeds up the t-SNE algorithm, achieving up to 30-fold acceleration in processing times for datasets with sizes on the order of 1 million points when embedding them in two dimensions. Such efficiency gains make it feasible to apply t-SNE to datasets consisting of millions of data points, far exceeding the capabilities of previous methods.

These improvements have direct implications for fields such as bioinformatics, where the analysis of extensive high-dimensional datasets from single-cell RNA-sequencing (scRNA-seq) is critical. Facilitating the visualization of such datasets without substantial computational resources expands the accessibility of t-SNE analyses beyond specialized settings.

Theoretical and Practical Implications

From a theoretical standpoint, the integration of FFT into gradient descent computations represents a substantial contribution to the numerical optimization domain, showcasing the benefits of practical applications of mathematical constructs like Fourier transforms.

Practically, FIt-SNE paves the way for more inclusive exploration of high-dimensional data by lowering computational barriers, fostering broader adoption across diverse scientific disciplines requiring large-scale data analysis. Furthermore, the insights on likelihood sampling could influence future theoretical exploration into clustering and manifold learning, potentially guiding algorithmic developments beyond the field of dimensionality reduction.

Future Directions

Future developments may include further refining these methods to accommodate real-time streaming data and exploring adaptive techniques that dynamically adjust the trade-off between accuracy and computational cost based on the user’s context and dataset characteristics. Additionally, advancements can leverage this work for more sophisticated embedding techniques that incorporate additional constraints or structures relevant to specialized data analysis domains.

By innovatively overcoming inherent scalability issues, this paper accentuates ongoing advancements in the domain of data visualization and dimensionality reduction, setting a precedent for future research and application development.