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Towards K-means-friendly Spaces: Simultaneous Deep Learning and Clustering

Bo Yang, Xiao Fu, Nicholas D. Sidiropoulos, Mingyi Hong

arXiv:1610.04794v2cs.LG

TL;DR

Separate DR and clustering pipelines, and prior joint methods based on linear transformations, may not capture complex data-generating processes or avoid collapsed representations. The paper jointly learns a DNN-based DR mapping, reconstruction, and K-means structure, using a scalable optimization procedure. Experiments on synthetic and real datasets report visible improvement over the respective state of the art across all evaluated datasets.

  • Problem

    Prior joint DR-and-clustering methods use linear transformations, while directly coupling DNN outputs to clustering can admit trivial collapsed representations.

  • Method

    The paper jointly learns a DNN-based dimensionality-reduction mapping with K-means structure, reconstruction through a decoding network, and an alternating optimization procedure.

  • Results

    Experiments on synthetic and real datasets report evidently visible improvement over the respective state of the art for all evaluated datasets.

  • Takeaways & Limitations

    The approach learns latent representations intended to make K-means suitable after high-dimensional data have undergone complex nonlinear transformations.

  • Takeaways & Limitations

    The paper concentrates on its proposed DCN as a proof of concept rather than exhausting combinations of network structures and clustering criteria.

Abstract

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Most learning approaches treat dimensionality reduction (DR) and clustering separately (i.e., sequentially), but recent research has shown that optimizing the two tasks jointly can substantially improve the performance of both. The premise behind the latter genre is that the data samples are obtained via linear transformation of latent representations that are easy to cluster; but in practice, the transformation from the latent space to the data can be more complicated. In this work, we assume that this transformation is an unknown and possibly nonlinear function. To recover the `clustering-friendly' latent representations and to better cluster the data, we propose a joint DR and K-means clustering approach in which DR is accomplished via learning a deep neural network (DNN). The motivation is to keep the advantages of jointly optimizing the two tasks, while exploiting the deep neural network's ability to approximate any nonlinear function. This way, the proposed approach can work well for a broad class of generative models. Towards this end, we carefully design the DNN structure and the associated joint optimization criterion, and propose an effective and scalable algorithm to handle the formulated optimization problem. Experiments using different real datasets are employed to showcase the effectiveness of the proposed approach.

1. Introduction

The paper proposes jointly learning dimensionality reduction and K-means clustering with a deep neural network, targeting latent representations that remain cluster-friendly under nonlinear data transformations. Its criterion combines reconstruction and cluster-structure regularization, and experiments report visible improvements over prior methods.

  • Motivation and contribution: Joint DR and K-means learning is proposed to produce low-dimensional representations suited to clustering rather than treating DR as a separate preprocessing stage.The approach replaces a linear DR model with a learned DNN to model more complex transformations.
  • Experiments: Synthetic experiments transform well-separated 2-D clusters into 100-D data through a complex nonlinear mapping, then compare recovered 2-D representations across several DR and clustering methods.The setup tests whether methods can recover a space suitable for K-means after nonlinear distortion.
  • Optimization criterion: The objective combines dimensionality reduction, data reconstruction, and cluster-structure-promoting regularization.A decoding network reconstructs the input and helps avoid trivial solutions.
  • Optimization procedure: The proposed optimization package uses empirically effective initialization and an alternating stochastic-gradient algorithm designed for nonlinear layers and K-means integer constraints.The resulting algorithm is described as simple, online-capable, and scalable.
  • Experiments: Experiments on synthetic and real datasets report evidently visible improvement over the respective state of the art for all evaluated datasets.The paper also provides experiment code for reproducibility.

2. Background and Related Works

The background contrasts conventional DR-plus-clustering pipelines and linear latent-variable models with joint DNN-based approaches. It also identifies trivial collapsed representations as a central problem in directly coupling a DNN to a clustering loss.

  • Conventional DR and clustering: K-means is effective when samples are evenly distributed around centroids, whereas high-dimensional data are often not K-means-friendly and can benefit from DR before clustering.Classic DR methods such as PCA and NMF learn lower-dimensional representations before K-means.
  • Linear joint models: Earlier joint DR-and-clustering work models data as xi = W hi, assuming clusters are well separated in a lower-dimensional latent domain but distorted by a linear transformation.Its objective balances data fidelity and latent cluster structure using regularization to prevent trivial solutions.
  • Motivation for nonlinear models: The paper argues that the linear model X ≈ W H may be oversimplified because real data-generating processes can involve more complex nonlinear transformations.This motivates using DNNs while retaining the joint DR-and-clustering idea.
  • Problem with direct joint deep clustering: Connecting a clustering module directly to a DNN output can yield the global optimum f(xi; W) = 0 with zero clustering loss.Other trivial solutions can map arbitrary samples into tight clusters without respecting the original data.
  • Problem with direct joint deep clustering: The problematic direct-joint structure lacks a provision for reconstructing or otherwise respecting the input samples, as illustrated for DCN without reconstruction and DEC.This motivates adding reconstruction to the proposed architecture.

3. Proposed Formulation

The proposed framework jointly learns a nonlinear dimensionality-reduction mapping and K-means-friendly latent representations. Its reconstruction-aware objective and encoder-decoder architecture are designed to prevent trivial solutions while supporting flexible network and clustering choices.

  • Proposed Formulation: The method uses a DNN to learn a nonlinear mapping from observable data to clustering-friendly latent representations.DNNs are chosen because they can approximate continuous mappings with a reasonable number of parameters.
  • Proposed Formulation: The decoding network reconstructs the input from bottleneck features, helping prevent trivial low-dimensional representations such as all-zero vectors.K-means is performed at the bottleneck, with encoding layers before it and decoding layers after it.
  • Proposed Formulation: The joint objective combines dimensionality reduction, data reconstruction, and cluster-structure-promoting regularization.The reconstruction loss uses least-squares fitting, while λ balances reconstruction against finding K-means-friendly latent representations.
  • Proposed Formulation: The framework is flexible enough to accommodate other network architectures and clustering criteria, including convolutional networks, K-subspace, and soft K-means.The paper focuses on DCN as a proof of concept rather than exhaustively testing combinations.

4. Optimization Procedure

The optimization procedure addresses the non-convex joint problem through pre-training and alternating stochastic updates. It updates network parameters, assignments, and centroids in separate blocks, enabling online and scalable implementation.

  • Optimization Procedure: The joint objective has non-convex costs and constraints, motivating a pragmatic procedure with initialization and alternating optimization.The method is designed to handle scalability as well as optimization difficulty.
  • Initialization: Layer-wise autoencoder pre-training initializes the encoder and decoder parameters before joint optimization.The authors report that this pre-training is important for the completely unsupervised DCN setting.
  • Alternating Updates: Alternating optimization updates one of the centroids, assignments, or network parameters while holding the other two variable sets fixed.Network updates use back-propagation-based stochastic gradient methods, while assignments are updated for incoming samples.
  • Centroid Updates: Centroid updates use adaptive learning rates based on cluster assignment counts to preserve balance in online learning.Clusters with many assigned members are updated more conservatively, while others are updated more aggressively.
  • Algorithm: Algorithm 1 cycles through network, assignment, and centroid updates, and supports mini-batch stochastic gradient descent and batch normalization.The resulting procedure can be implemented completely online and is described as very scalable.

5. Experiments

The experiments evaluate DCN on synthetic and real-world data using an implementation built with Theano. This section introduces the experimental evaluation rather than reporting a specific result.

  • Experiments: The evaluation uses both synthetic and real-world datasets to assess DCN’s effectiveness.The implementation uses Theano’s deep learning toolbox.

5.1. Synthetic-Data Demonstration

The synthetic demonstration tests whether DCN can recover a clustering-friendly latent space when observed data arise from a difficult nonlinear transformation. DCN succeeds where the compared dimensionality-reduction methods fail, and similar behavior appears under additional generative models.

  • Synthetic-Data Demonstration: Recovering the latent K-means-friendly domain is difficult because the observed R100 data are produced through a nonlinear transformation.The latent representations are denoted hi, while xi denotes the observed samples.
  • Synthetic-Data Demonstration: The synthetic setup assumes four clusters in an unobserved two-dimensional latent domain, with each cluster containing 2,500 samples.The observed samples lie in R100 and are generated through a nonlinear transformation involving Gaussian matrices and a sigmoid function.
  • Synthetic-Data Demonstration: All compared dimensionality-reduction methods except DCN fail to map the observations to a two-dimensional space suitable for K-means.The comparison includes NMF, LLE, Laplacian eigenmap, DEC, and DCN without reconstruction.
  • Synthetic-Data Demonstration: Additional simulations with different generative models produce similar results, illustrating DCN’s ability to recover clustering-friendly structure across nonlinear models.These additional simulations are reported in the supplementary materials.

5.2. Real-Data Validation

Experiments on RCV1-v2, 20Newsgroup, and MNIST evaluate DCN against clustering and dimensionality-reduction baselines using standard clustering metrics. Across these datasets, DCN improves over relevant baselines, including sequential SAE+KM and methods without DNNs.

  • Experimental setup: The evaluation compares DCN with classic clustering, factorization, autoencoder, joint NMF, and deep embedded clustering baselines using NMI, ARI, and ACC.Some baselines are omitted from large datasets because they are not scalable enough.
  • RCV1-v2: RCV1-v2 experiments progressively increase from 4 to 20 clusters, creating increasingly unbalanced and difficult test cases.The corpus subset contains 365,968 documents from 20 topics, and cluster sizes are highly uneven.
  • RCV1-v2: Across the RCV1-v2 cases, DCN clearly improves over SAE+KM, KM, and XRAY on the reported evaluation metrics.DCN outperforms the two-stage SAE+KM approach in almost all cases and across all evaluation metrics.
  • Training behavior: NMI, ARI, and ACC show a clear ascending trend across training epochs for the proposed algorithm on the RCV1 4-clusters visualization.The observation is empirical; sufficient conditions guaranteeing such improvement are left for future work.
  • 20Newsgroup: On 20Newsgroup, SAE+KM substantially improves ARI and ACC over non-DNN methods, while DCN performs even better with joint DR and clustering.The result supports using K-means regularization to discover a clustering-friendly space.
  • Raw MNIST: On raw MNIST, DCN is approximately on par with DEC, and both outperform the other evaluated methods by a large margin.The balanced MNIST setting is favorable to DEC, whereas DCN also performs well on the unbalanced RCV1-v2 data.
  • Pre-processed MNIST: On pre-processed MNIST, DCN achieves the best performance among the evaluated algorithms and is proposed as a replacement for K-means components in related pipelines.The authors suggest this may benefit methods such as spectral clustering and sparse subspace clustering.
  • Parameter sensitivity: Performance is generally insensitive to the exact reconstruction–clustering trade-off parameter λ, although performance degrades when λ becomes inappropriately large.The parameter balances reconstruction error and clustering regularization.

6. Conclusion

The paper proposes joint dimensionality reduction and K-means clustering, using a deep neural network to map high-dimensional data into a clustering-friendly latent space.

  • The approach learns a deep neural network for dimensionality reduction jointly with K-means clustering.
  • The network structure is designed to avoid trivial and meaningless latent representations.
  • Synthetic and real-data experiments show the algorithm is effective across varied datasets.

1. Additional Synthetic-Data Experiments

Additional synthetic experiments test DCN under more complicated nonlinear generative models and find that it recovers clustering-friendly latent spaces.

  • The experiments evaluate DCN under additional complicated nonlinear transformations.
  • Elementwise squaring is added to sigmoid-transformed features, making the generation process more complex.
  • DCN recovers a 2-D K-means-friendly space very well under the squared transformation, while the other methods fail.
  • Under another generative model, DCN produces very clear clusters in the recovered 2-D space.
  • Across these and the main-text synthetic experiments, DCN outputs clustering-friendly latent representations under varied complicated nonlinear models.

2. Additional Real-Data Experiments

Additional real-data experiments examine DCN on Pendigits and MNIST settings, including feature learning and preprocessed inputs.

  • The Pendigits experiment evaluates DCN on 10,992 samples representing handwritten digits as length-16 coordinate vectors.
  • The supplementary figures illustrate generated and recovered 2-D representations for nonlinear transformations involving sigmoid functions, squaring, and tanh.
  • DCN is also presented as a feature-learning method that can reduce the dimension of unseen testing data after training on labeled samples.
  • A 5-fold MNIST cross-validation experiment uses 80/20 training/testing splits to assess DCN as a representation learner.
  • Training and testing stages of DCN produce similar NMI, ARI, and ACC results on MNIST.

3. Detailed Settings of Real-Data Experiments

The real-data experiments use a two-stage, mirrored encoding-decoding network and specify optimization and architecture settings for each dataset.

  • The algorithm has separate pre-training and main-algorithm stages with different parameter settings.
  • Optimization uses mini-batch stochastic gradient descent with Nesterov-type acceleration, momentum, and batch normalization.
  • The network uses mirrored forward encoding and decoding parts, with width and depth selected according to the data and task.
  • The experiments provide dataset-specific parameter settings for RCV1-v2, 20Newsgroup, raw MNIST, preprocessed MNIST, and Pendigits.

4. More Discussions

The discussion examines how training choices affect clustering and reports safeguards for experimental variability. It frames the method as a proof of concept while identifying theoretical and optimization issues for future work.

  • Training effects: SAE training without K-means regularization can worsen clustering performance compared with using only pre-training initialization.Figure 3 tracks clustering-performance indexes across SAE epochs and supports the motivation for structure-enhancing regularization.
  • Experimental protocol: The reported experiments average multiple runs to reduce effects from algorithmic randomness, with ten runs for smaller datasets and five for RCV-v2.DEC results in Table 1 come from a single run.
  • Future work: The authors characterize joint DNN learning and clustering as a proof of concept and leave theory-backed parameter selection and convergence-guaranteed optimization for future work.
  • Training effects: Clustering performance degrades when training uses only the reconstruction-error term, unlike the proposed DCN model, where it improves.
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