# Semi-Supervised Learning via Compact Latent Space Clustering

international conference on machine learning, pp. 2464-2473, 2018.

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Generative adversarial networksEuropean Research Councilcompact clustercost functionvariational auto-encodersMore(16+)

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Abstract:

We present a novel cost function for semi-supervised learning of neural networks that encourages compact clustering of the latent space to facilitate separation. The key idea is to dynamically create a graph over embeddings of labeled and unlabeled samples of a training batch to capture underlying structure in feature space, and use label...More

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Introduction

- Semi-supervised learning (SSL) addresses the problem of learning a model by effectively leveraging both labeled and unlabeled data (Chapelle et al, 2006).
- SSL is effective when it results in a model that generalizes better than a model learned from labeled data only.
- Let X be a sample space with data points and Y the set of labels.
- Let DL ⊆ X × Y be a set of labeled data points, and let DU ⊆ X be a set of unlabeled data.
- Leveraging the ample unlabeled data allows capturing more faithfully the structure of data

Highlights

- Semi-supervised learning (SSL) addresses the problem of learning a model by effectively leveraging both labeled and unlabeled data (Chapelle et al, 2006)
- We study the dynamics of Compact Clustering via Label Propagation (S = 10) on two-circles (Fig. 2), when a single labeled example is given per class
- Compact Clustering via Label Propagation consistently improves performance over standard supervision even when all labels in the training set are used for supervision, indicating that Compact Clustering via Label Propagation could be used as a latent space regularizer in fully supervised systems
- We emphasize that our method consists of the computation of a single cost function and does not require additional network components, such as the generators required for variational auto-encoders and Generative adversarial networks, or the density estimator PixelCNN++ used in Dai et al (2017)
- We have presented a novel regularization technique for supervised learning, based on the idea of forming compact clusters in the latent space of a neural network while preserving existing clusters during optimization
- We showed that our approach is effective in leveraging unlabeled samples via empirical evaluation on three widely used image classification benchmarks

Methods

- In this work the authors take the labeling function f (x; θ) to be a multi-layer neural network.
- This model can be decomposed into a feature extractor z(x; θz) ∈ Z parametrized by θz, and a classifier g(z(x; θz); θg) with parameters θg.
- The authors argue that classification is improved whenever data from each class form compact, well separated clusters in feature space Z.
- The authors introduce a regularizer (Section 3.2) that 1) encourages compact clustering according to propagated labels and 2) avoids disturbing existing clusters during optimization (Fig. 1)

Results

- Performance of the method in comparison to recent SSL approaches that use similar experimental settings are reported in Table 1.
- CCLP consistently improves performance over standard supervision even when all labels in the training set are used for supervision, indicating that CCLP could be used as a latent space regularizer in fully supervised systems.
- In the latter settings, CCLP offers greater improvement over the corresponding baselines than the most recent perturbation-based method, mean teacher (Tarvainen & Valpola, 2017).
- The compact clustering that the method encourages is orthogonal to previous approaches and could boost their performance further

Conclusion

- The authors have presented a novel regularization technique for SSL, based on the idea of forming compact clusters in the latent space of a neural network while preserving existing clusters during optimization.
- This is enabled by dynamically constructing a graph in latent space at each SGD iteration and propagating labels to estimate the manifold’s structure, which the authors regularize.
- Analyzing further the properties of a compactly clustered latent space, as well as applying the approach to larger benchmarks and the task of semantic segmentation is interesting future work

Summary

## Introduction:

Semi-supervised learning (SSL) addresses the problem of learning a model by effectively leveraging both labeled and unlabeled data (Chapelle et al, 2006).- SSL is effective when it results in a model that generalizes better than a model learned from labeled data only.
- Let X be a sample space with data points and Y the set of labels.
- Let DL ⊆ X × Y be a set of labeled data points, and let DU ⊆ X be a set of unlabeled data.
- Leveraging the ample unlabeled data allows capturing more faithfully the structure of data
## Methods:

In this work the authors take the labeling function f (x; θ) to be a multi-layer neural network.- This model can be decomposed into a feature extractor z(x; θz) ∈ Z parametrized by θz, and a classifier g(z(x; θz); θg) with parameters θg.
- The authors argue that classification is improved whenever data from each class form compact, well separated clusters in feature space Z.
- The authors introduce a regularizer (Section 3.2) that 1) encourages compact clustering according to propagated labels and 2) avoids disturbing existing clusters during optimization (Fig. 1)
## Results:

Performance of the method in comparison to recent SSL approaches that use similar experimental settings are reported in Table 1.- CCLP consistently improves performance over standard supervision even when all labels in the training set are used for supervision, indicating that CCLP could be used as a latent space regularizer in fully supervised systems.
- In the latter settings, CCLP offers greater improvement over the corresponding baselines than the most recent perturbation-based method, mean teacher (Tarvainen & Valpola, 2017).
- The compact clustering that the method encourages is orthogonal to previous approaches and could boost their performance further
## Conclusion:

The authors have presented a novel regularization technique for SSL, based on the idea of forming compact clusters in the latent space of a neural network while preserving existing clusters during optimization.- This is enabled by dynamically constructing a graph in latent space at each SGD iteration and propagating labels to estimate the manifold’s structure, which the authors regularize.
- Analyzing further the properties of a compactly clustered latent space, as well as applying the approach to larger benchmarks and the task of semantic segmentation is interesting future work

- Table1: Performance of CCLP compared to contemporary SSL methods on common benchmarks, when limited or all available labelled data is used as DL for training. Also shown is performance of the corresponding baseline with standard supervision (no SS). Error rate is shown as (mean ± st.dev.). Only results obtained without augmentation are shown. Methods in the lower part used larger classifiers

Related work

- The great potential and practical implications of utilizing unlabeled data has resulted in a large body of research on SSL. The techniques can be broadly categorized as follows.

2.1. Graph-Based Methods

These methods operate over an input graph with adjacency matrix A, where element Aij is the similarity between samples xi, xj ∈ DL ∪ DU . Similarity can be based on Euclidean distance (Zhu & Ghahramani, 2002) or other, sometimes task-specific metrics (Weston et al, 2012). Transductive inference for the graph’s unlabeled nodes is done based on the smoothness assumption, that nearby samples should have similar class posteriors. Label propagation (LP) (Zhu & Ghahramani, 2002) iteratively propagates the class posterior of each node to neighbors, faster through high density regions, until a global equilibrium is reached. Zhu et al (2003) showed that for binary classification one arrives at the same solution by minimizing the energy:

Funding

- This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 757173, project MIRA, ERC-2017-STG)
- KK is also supported by the President’s PhD Scholarship of Imperial College London
- DC is supported by CAPES, Ministry of Education, Brazil (BEX 1500/15-05)
- LLF is funded through EPSRC Healthcare Impact Partnerships grant (EP/P023509/1)
- IW is supported by the Natural Environment Research Council (NERC)

Reference

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