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Improved Distributed Principal Component Analysis

Maria-Florina Balcan, Vandana Kanchanapally, Yingyu Liang, David Woodruff

arXiv:1408.5823v5cs.LG

TL;DR

Distributed PCA must balance communication and computational efficiency while supporting downstream k-means and related fitting problems. The paper introduces improved communication and randomized computational techniques, including high-probability sparse embeddings. These methods achieve orders-of-magnitude speedups while preserving communication and causing only negligible solution-quality degradation in experiments.

  • Problem

    Distributed PCA seeks efficient communication and computation for finding low-dimensional structure across partitioned data and enabling approximate k-means and related ℓ2-fitting problems.

  • Method

    The paper improves distributed PCA through close-projection communication reductions, oblivious subspace embeddings, randomized approximate SVD, and probability boosting for sparse embeddings.

  • Results

    Orders-of-magnitude computational speedups preserve communication while causing only negligible degradation in solution quality on real-world data.

  • Takeaways & Limitations

    The resulting techniques improve distributed PCA for k-means and related ℓ2-fitting applications, while the probability-boosting transformation may be independently useful.

  • Takeaways & Limitations

    The main analysis assumes centered data; uncentered data requires a rank-one modification to the algorithms.

Abstract

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We study the distributed computing setting in which there are multiple servers, each holding a set of points, who wish to compute functions on the union of their point sets. A key task in this setting is Principal Component Analysis (PCA), in which the servers would like to compute a low dimensional subspace capturing as much of the variance of the union of their point sets as possible. Given a procedure for approximate PCA, one can use it to approximately solve $\ell_2$-error fitting problems such as $k$-means clustering and subspace clustering. The essential properties of an approximate distributed PCA algorithm are its communication cost and computational efficiency for a given desired accuracy in downstream applications. We give new algorithms and analyses for distributed PCA which lead to improved communication and computational costs for $k$-means clustering and related problems. Our empirical study on real world data shows a speedup of orders of magnitude, preserving communication with only a negligible degradation in solution quality. Some of these techniques we develop, such as a general transformation from a constant success probability subspace embedding to a high success probability subspace embedding with a dimension and sparsity independent of the success probability, may be of independent interest.

1 Introduction

The paper develops distributed PCA methods that improve communication and computation for downstream k-means and related ℓ2-fitting problems. Its randomized speedups preserve communication and deliver orders-of-magnitude computational savings with negligible solution-quality degradation in experiments.

  • Motivation: Distributed PCA supports low-dimensional variance capture and preprocessing for k-means, NNMF, LDA, and related tasks.PCA finds an r-dimensional subspace capturing as much data variance as possible and can reduce dimension before downstream learning.
  • Baseline Distributed PCA: The baseline disPCA protocol sends each server’s top O(r/ϵ) principal components to a coordinator, which computes the global top r components.The coordinator stacks the transmitted local components and performs PCA on the resulting matrix.
  • Improved Communication: O(skd/ϵ2) communication suffices to compute O(k/ϵ2) vectors preserving k-means cost within a (1 + ϵ) factor.This improves the previous O(skd/ϵ3) communication by a factor of 1/ϵ through the close projection property.
  • Improved Computation: Randomized embeddings and approximate SVD replace costly local and coordinator SVDs while preserving communication and sparsity.Oblivious subspace embeddings can be computed in nnz(Pi) time, with no more nonzeros in HiPi than in Pi.
  • Improved Computation: Independent repetitions boost constant-probability embeddings to success probability 1 − δ without increasing dimension and sparsity by a factor of the server count.The procedure applies O(log 1/δ) embeddings and uses cross-validation to identify a successful one.
  • Empirical Results: Experiments report orders-of-magnitude computational savings, preserved communication, and only negligible degradation in solution quality.On BOWpubmed, normal PCA did not finish in time, whereas the speedup versions produced good results in reasonable time.

2 Preliminaries

The preliminaries define the distributed data model, PCA and related ℓ2-error fitting problems, and the centered-data assumption used throughout the paper. They also introduce the SVD-based distributed PCA algorithmic framework.

  • Communication Model: The global matrix P is partitioned across s servers, each holding a local matrix Pi, with communication through a central coordinator.Each local matrix contains ni points in d dimensions, and the global data is formed by concatenating the local matrices.
  • Assumptions: The paper assumes the data points are centered to have zero mean; uncentered data requires a rank-one modification.The communication and computation costs of that modification are dominated by the other algorithmic steps.
  • Distributed PCA: The distributed disPCA protocol sends each server’s top O(r/ϵ) local principal components to the coordinator for a global top-r computation.This protocol provides a relative-error approximation to the PCA problem.
  • PCA and Fitting Problems: The framework also includes k-means and constrained variants such as NNMF and LDA as related ℓ2-error fitting problems.In k-means, the centers are points, corresponding to r = 0 in the subspace-clustering formulation.
  • PCA and Fitting Problems: PCA is the k = 1 special case of subspace clustering, and its optimal r-dimensional subspace is spanned by the top r eigenvectors or right singular vectors.These vectors can be computed using the SVD of P.

3 Tradeoff between Communication and Solution Quality

disPCA reduces distributed PCA and downstream ℓ2-fitting costs by communicating local SVD summaries, while close projection preserves solution quality for low-dimensional fitting tasks. The resulting guarantees trade larger projected dimensions for approximation accuracy and support distributed k-means with controlled communication and error.

  • 3 Tradeoff between Communication and Solution Quality: Algorithm disPCA communicates local singular-vector summaries, stacks them at a coordinator, and computes global directions from the resulting matrix.The local stage uses truncated SVDs; the global stage performs SVD on the concatenated summaries.
  • 3 Tradeoff between Communication and Solution Quality: With t1 ≥ r + ⌈4r/ϵ⌉ − 1 and t2 = r, disPCA yields a (1 + ϵ)-approximation for distributed PCA with communication O(srd).The approximation follows because local SVD truncations can serve as a proxy for the full distributed data.
  • 3.1 Guarantees for Distributed ℓ2-Error Fitting: Theorem 3 uses t1 = t2 = O(rk/ϵ2) to ensure projected data preserves every r-subspace k-clustering cost up to the stated approximation guarantee.This close projection guarantee applies to any set of k centers in r-Subspace k-Clustering.
  • 3.1 Guarantees for Distributed ℓ2-Error Fitting: The projected-data guarantee introduces only a small multiplicative approximation factor of (1 + O(ϵ)), although the resulting vectors are not optimal global principal components.The vectors instead satisfy the weaker close projection property needed for downstream fitting guarantees.
  • 3.1 Guarantees for Distributed ℓ2-Error Fitting: For distributed k-means, Algorithm 2 outputs a (1 + ϵ)2α-approximate solution with total communication bounded by the theorem's stated O(sk… expression.It first runs disPCA with t1 = t2 = O(k/ϵ2), then applies a distributed clustering algorithm to the projected data.

4 Fast Distributed PCA

Fast distributed PCA combines local subspace embeddings, high-probability embedding selection, and randomized SVD to reduce computation while preserving guarantees for ℓ2-error fitting.

  • Subspace Embeddings: Subspace embeddings reduce distributed PCA time while retaining guarantees sufficient for ℓ2-error fitting.Each server embeds its local data before running distributed PCA.
  • High-Probability Embeddings: Sparse embeddings avoid the factor-s increase in embedding dimension that naive high-probability amplification would require.The technique is designed for embeddings with one nonzero entry per column.
  • High-Probability Embeddings: O(log 1/δ) independent constant-success embeddings and cross-validation produce a selected embedding with success probability 1−δ.The selection compares singular values across independently embedded matrices.
  • Randomized SVD: Randomized SVD probes the row space, computes SVD on a smaller projected matrix, and uses power iterations to emphasize leading singular vectors.This replaces exact SVD in the fast distributed PCA algorithm.
  • Fast Distributed PCA: Algorithm 3 applies local embeddings and randomized distributed PCA to obtain a subspace suitable for low-dimensional ℓ2-error fitting.Its analysis transfers close-projection guarantees through embedding and randomized-SVD errors.

5 Experiments

Experiments on real-world datasets evaluate fast distributed PCA for low-rank approximation, k-means, and PCR, finding large runtime reductions with nearly unchanged solution quality.

  • Experimental Setup: The experiments cover rank-r approximation, k-means clustering, and principal component regression using real-world UCI datasets.Costs are reported relative to corresponding solutions computed from global data.
  • Results: 10 to 100× faster distributed PCA is achieved on NewsGroups, while BOWpubmed becomes feasible when normal PCA takes too long.The speedup versions produce good results in reasonable time on the large dataset.
  • Results: Distributed solutions for k-means and PCR are nearly as good as solutions computed directly on global data.Figures 3 and 4 report these comparisons.

A.1 Proof of Lemma 1

The proof establishes a close-projection bound for truncated SVD, using singular-value truncation and norm inequalities to compare projection errors.

  • Lemma 7: Lemma 7 bounds the projection error after retaining the first t singular directions for any orthonormal comparison matrix.The condition requires t to exceed a rank- and accuracy-dependent threshold.
  • Proof: The proof rewrites the truncated matrix through its SVD and applies orthonormality and spectral-norm consistency.These identities yield the claimed inequality.
  • Proof: Lemma 1 follows by applying Lemma 7 to orthonormal matrices and using the Eckart–Young property of SVD truncation.The argument compares approximation distances for the original and truncated matrices.

A.2 Proof of Theorem 2

Theorem 2 analyzes distributed PCA by comparing distances for the original and locally compressed data, with a data-dependent refinement reducing the required local rank in practice.

  • Theorem 2: Theorem 2 guarantees distributed PCA accuracy when t1 ≥ r + ⌈4r/ϵ⌉−1 and t2 = r.The output is the r-dimensional subspace V(r).
  • Proof: The proof decomposes the difference between global and compressed-data projection errors into three terms and bounds them separately.The optimality of V(r) for the compressed data makes one term nonpositive.
  • Data-Dependent Refinement: A refined data-dependent bound replaces the worst-case local rank threshold with τ(Pi, r, ϵ).This threshold can be much smaller than O(r/ϵ) in practice.
  • Data-Dependent Refinement: With t1 ≥ maxi τ(Pi, r, ϵ) and t2 = r, the communication cost is O(sd maxi τ(Pi, r, ϵ)) words.The bound adapts communication to the largest local data-dependent threshold.

B.1 Proof of Lemma 4

Lemma 4 bounds how local and global projections differ when disPCA uses truncated singular vectors, establishing a close-projection property for orthonormal test subspaces.

  • The proof compares two successive projections: each local data matrix is projected to its local subspace, then the concatenated result is projected globally.The intermediate matrices bP and P formalize these local and global projections.
  • When t1 = t2 ≥ k + ⌈8k/ϵ⌉−1, Lemma 4 applies to every d × k orthonormal matrix X.The lemma is the parameterized statement underlying the projection guarantees.
  • The analysis bounds errors for the intermediate projections and transfers those bounds to the concatenated data using Lemma 7 and a weak triangle inequality.Separate bounds are combined across the proof’s numbered claims.
  • The resulting close-projection guarantee is obtained by bounding d2(bP, LX) relative to d2(P, LX) and invoking Claim 1.The proof concludes after relating the intermediate and final projected data.
  • The second statement follows from the preceding inequalities and Claim 1.

B.2 Proof of Theorem 3

Theorem 3 extends the distributed PCA guarantee from k-means to general ℓ2-error geometric fitting, using a weak triangle inequality that increases the required projection dimension.

  • A weak triangle inequality and Pythagorean-theorem arguments control the component-wise errors needed for the theorem.The proof repeatedly applies these inequalities to compare projected and original costs.
  • For k-means, the projection parameter must satisfy t1 = t2 = O(k/ϵ2), because the weak triangle inequality loses a factor of 1/ϵ.
  • The theorem’s proof follows the distributed-setting argument of Feldman et al. with a slight modification.
  • The proof decomposes the costs of P and its projection into distance to the subspace LX and distance from that projection to the centers.This decomposition enables separate comparison of the two contributions.
  • The proof first handles k-means and then extends the same argument to general ℓ2-error fitting problems.The theorem is stated for r-Subspace k-Clustering and the proof follows the k-means case.

C.1 Proofs for Subspace Embedding

The subspace-embedding proofs establish sparse embedding guarantees, boost constant success probability to 1−δ, and analyze randomized SVD costs and accuracy.

  • A sparse embedding H preserves all vectors’ norms within 1 ± ϵ with probability at least 99/100 and can be computed in O(nnz(A)) time.The embedding dimension is given as O(d2/ϵ2) in the supplied statement.
  • Algorithm 5 boosts success probability to at least 1−δ by constructing O(log 1/δ) independent embeddings and selecting one through pairwise verification.The verification procedure identifies an embedding that agrees with a majority of the others.
  • The boosted procedure runs Step 3 only a constant expected number of times and has expected time O(d3r2/ϵ2).Randomly choosing a candidate makes the expected number of trials constant.
  • Randomized SVD returns a rank-t projection with probability at least 1−3e−t and runs in O(qtℓd + t2(ℓ + d)) time.The accuracy analysis supplies the high-probability guarantee used by the later distributed-PCA proof.

C.3 Proof of Theorem 6

Theorem 6 shows that replacing exact local SVD with subspace embeddings and randomized SVD preserves the distributed PCA fitting guarantee with high probability while controlling communication and runtime.

  • The input to disPCA is TP, where T is block diagonal with the local embeddings, and the projected output is T˜P.The embedding-based analysis compares this transformed projection with the desired projected data.
  • The proof first shows that randomized SVD approximately preserves the relevant properties, then shows that subspace embedding transfers them to the final projection.These two approximation steps replace exact SVD in disPCA.
  • With probability at least 1−δ, Algorithm 3 produces a projected dataset satisfying the ℓ2-error fitting guarantee for every set of k points L.The theorem sets each local embedding failure probability to δ′ = δ/2s.
  • The total communication is O(skd/ϵ2), while the total running time follows from local embedding, randomized-SVD, and coordinator costs.Each node sends only its truncated singular values and right singular vectors.
  • Correctness follows from the proof of Theorem 3 after replacing Lemma 4 with Lemma 17.
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