American Institute of Mathematical Sciences

Advanced
September  2019, 1(3): 271-291. doi: 10.3934/fods.2019012

Learning by active nonlinear diffusion

Mauro Maggioni 1, and  James M. Murphy 2,,

1. 

Department of Mathematics, Department of Applied Mathematics and Statistics, Mathematical Institute of Data Sciences, Institute of Data Intensive Engineering and Science, Johns Hopkins University, Baltimore, MD 21218, USA
2. 

Department of Mathematics, Tufts University, Medford, MA 02155, USA

* Corresponding author: James M. Murphy

Published  August 2019

Fund Project: This research is supported by NSF-DMS-125012, NSF-DMS-1724979, NSF-DMS-1708602, NSF-ATD-1737984, AFOSR FA9550-17-1-0280, NSF-IIS-1546392, NSF-DMS 1912737, and NSF-DMS 1924513.

This article proposes an active learning method for high-dimensional data, based on intrinsic data geometries learned through diffusion processes on graphs. Diffusion distances are used to parametrize low-dimensional structures on the dataset, which allow for high-accuracy labelings with only a small number of carefully chosen training labels. The geometric structure of the data suggests regions that have homogeneous labels, as well as regions with high label complexity that should be queried for labels. The proposed method enjoys theoretical performance guarantees on a general geometric data model, in which clusters corresponding to semantically meaningful classes are permitted to have nonlinear geometries, high ambient dimensionality, and suffer from significant noise and outlier corruption. The proposed algorithm is implemented in a manner that is quasilinear in the number of unlabeled data points, and exhibits competitive empirical performance on synthetic datasets and real hyperspectral remote sensing images.

Keywords: Active learning, statistical learning, diffusion geometry, machine learning, spectral graph theory.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Mauro Maggioni, James M. Murphy. Learning by active nonlinear diffusion. Foundations of Data Science, 2019, 1 (3) : 271-291. doi: 10.3934/fods.2019012
References:
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show all references

References:
[1]

N. Acito, G. Corsini and M. Diani, An unsupervised algorithm for hyperspectral image segmentation based on the gaussian mixture model, in IEEE International Geoscience and Remote Sensing Symposium (IGARSS), 6 (2003), 3745–3747. doi: 10.1109/IGARSS.2003.1295256.

[2]

A. Anis, A. Gadde and A. Ortega, Towards a sampling theorem for signals on arbitrary graphs, in 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2014, 3864–3868. doi: 10.1109/ICASSP.2014.6854325.

[3]

A. Anis, A. Gadde and A. Ortega, Efficient sampling set selection for bandlimited graph signals using graph spectral proxies, IEEE Transactions on Signal Processing, 64 (2016), 3775-3789.  doi: 10.1109/TSP.2016.2546233.

[4]

A. Anis, A. E. Gamal, S. Avestimehr and A. Ortega, Asymptotic justification of bandlimited interpolation of graph signals for semi-supervised learning, in 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2015, 5461–5465. doi: 10.1109/ICASSP.2015.7179015.

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E. Arias-Castro, G. Lerman and T. Zhang, Spectral clustering based on local PCA, Journal of Machine Learning Research, 18 (2017), 1-57. 

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M.-F. Balcan, A. Beygelzimer and J. Langford, Agnostic active learning, Journal of Computer and System Sciences, 75 (2009), 78-89.  doi: 10.1016/j.jcss.2008.07.003.

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N. Cahill, W. Czaja and D. Messinger, Schroedinger eigenmaps with nondiagonal potentials for spatial-spectral clustering of hyperspectral imagery, in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery XX, vol. 9088, International Society for Optics and Photonics, 2014, 908804.

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[14]

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[15]

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J. Cohen, A coefficient of agreement for nominal scales, Educational and Psychological Measurement, 20 (1960), 37-46.  doi: 10.1177/001316446002000104.

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D. Cohn, L. Atlas and R. Ladner, Improving generalization with active learning, Machine Learning, 15 (1994), 201-221.  doi: 10.1007/BF00993277.

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R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.

[23]

R. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner and S. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 7426-7431. 

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S. Dasgupta, Two faces of active learning, Theoretical Computer Science, 412 (2011), 1767-1781.  doi: 10.1016/j.tcs.2010.12.054.

[25]

S. Dasgupta and D. Hsu, Hierarchical sampling for active learning, in Proceedings of the 25th International Conference on Machine Learning, ACM, 2008,208–215. doi: 10.1145/1390156.1390183.

[26]

S. Dasgupta, D. Hsu and C. Monteleoni, A general agnostic active learning algorithm, in Advances in neural information processing systems, 2008,353–360.

[27]

A. Esteva, B. Kuprel, R. Novoa, J. Ko, S. Swetter, H. Blau and S. Thrun, Dermatologist-level classification of skin cancer with deep neural networks, Nature, 542 (2017), 115-118.  doi: 10.1038/nature21056.

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J. Friedman, T. Hastie and R. Tibshirani, The Elements of Statistical Learning, vol. 1, Springer series in Statistics Springer, Berlin, 2001. doi: 10.1007/978-0-387-21606-5.

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N. Garcia Trillos, M. Gerlach, M. Hein and D. Slepcev, Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace–Beltrami operator, arXiv: 1801.10108.

[30]

N. Garcia Trillos, F. Hoffmann and B. Hosseini, Geometric structure of graph Laplacian embeddings, arXiv: 1901.10651.

[31]

M. Gavish and B. Nadler, Normalized cuts are approximately inverse exit times, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 757-772.  doi: 10.1137/110826928.

[32]

N. Gillis, D. Kuang and H. Park, Hierarchical clustering of hyperspectral images using rank-two nonnegative matrix factorization, IEEE Transactions on Geoscience and Remote Sensing, 53 (2015), 2066-2078.  doi: 10.1109/TGRS.2014.2352857.

[33]

J. Ham, Y. Chen, M. Crawford and J. Ghosh, Investigation of the random forest framework for classification of hyperspectral data, IEEE Transactions on Geoscience and Remote Sensing, 43 (2005), 492-501.  doi: 10.1109/TGRS.2004.842481.

[34]

S. Hanneke, Rates of convergence in active learning, The Annals of Statistics, 39 (2011), 333-361.  doi: 10.1214/10-AOS843.

[35]

A. Krizhevsky, I. Sutskever and G. Hinton, Imagenet classification with deep convolutional neural networks, Communications of the ACM, 60 (2017), 84-90.  doi: 10.1145/3065386.

[36]

S. Lafon and A. Lee, Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 1393-1403. 

[37]

J. Li, J. Bioucas-Dias and A. Plaza, Semisupervised hyperspectral image segmentation using multinomial logistic regression with active learning, IEEE Transactions on Geoscience and Remote Sensing, 48 (2010), 4085-4098.  doi: 10.1109/TGRS.2010.2060550.

[38]

J. Li, J. Bioucas-Dias and A. Plaza, Semisupervised hyperspectral image classification using soft sparse multinomial logistic regression, IEEE Geoscience and Remote Sensing Letters, 10 (2013), 318-322. 

[39]

A. Little, M. Maggioni and J. Murphy, Path-based spectral clustering: Guarantees, robustness to outliers, and fast algorithms, arXiv: 1712.06206.

[40]

M. Maggioni and J. Murphy, Learning by unsupervised nonlinear diffusion, arXiv: 1810.06702.

[41]

F. Melgani and L. Bruzzone, Classification of hyperspectral remote sensing images with support vector machines, IEEE Transactions on geoscience and remote sensing, 42 (2004), 1778-1790. 

[42]

D. Mixon, S. Villar and R. Ward, Clustering subgaussian mixtures by semidefinite programming, Information and Inference: A Journal of the IMA, 6 (2017), 389-415.  doi: 10.1093/imaiai/iax001.

[43]

J. Murphy and M. Maggioni, Iterative active learning with diffusion geometry for hyperspectral images, in 9th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), IEEE, 2018, 1–5. doi: 10.1109/WHISPERS.2018.8747033.

[44]

J. Murphy and M. Maggioni, Spectral-spatial diffusion geometry for hyperspectral image clustering, arXiv: 1902.05402.

[45]

J. Murphy and M. Maggioni, Unsupervised clustering and active learning of hyperspectral images with nonlinear diffusion, IEEE Transactions on Geoscience and Remote Sensing, 57 (2019), 1829-1845.  doi: 10.1109/TGRS.2018.2869723.

[46]

B. Nadler and M. Galun, Fundamental limitations of spectral clustering, in Advances in Neural Information Processing Systems, 2007, 1017–1024.

[47]

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Figure 1.  Data colored by class label. Both the data in (a) and (b) exhibit cluster structure, which can guide active learning in the case that the labels are constant on these clusters. Indeed, on the left, using a simple clustering algorithm such as $ K $-means suggests that only 3 labels are necessary to correctly label the entire dataset. For the data on the right, many more than three labels are necessary if $ K $-means is used for the underlying clustering, since the clusters are highly elongated and nonlinear. Indeed, $ K $-means will split the annular and elongated clusters. On the other hand, if pairwise comparisons are made with distances other than Euclidean distances, it may be possible that active learning achieves near perfect results with only 3 labels. The proposed active learning scheme gains robustness to class shape via diffusion geometry, and is suitable for data in both (a) and (b)
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Figure 2.  In (a) and (b), the values of $ \log_{10}( \mathcal{D}_{t}) $ are shown for synthetic geometric data from Figure 1 (b) for $ \log_{10}(t) = 2 $ and $ \log_{10}(t) = 9 $, respectively. We see that for small values of $ t $, the mode estimation incorrectly places the first three modes on the highly elongated cluster. For larger time values, the underlying random walk reaches a mesoscopic equilibrium and correct mode estimation is achieved. The emergence of mesoscopic equilibria is apparent in (c), (d), which show the matrix of diffusion distances at time $ \log_{10}(t) = 2 $ and $ \log_{10}(t) = 9 $, respectively. When $ \log_{10}(t) = 2 $, $ P^{t} $ has not mixed, and there are still substantial within-cluster distances. For $ \log_{10}(t) = 9 $, $ P^{t} $ has reached mesoscopic equilibria, so that within-cluster distances are quite small, yet between-cluster distances are still large [40]
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Figure 3.  Top row: Three different synthetic datasets in two dimensions are shown, categorized as geometric, bottleneck and Gaussian. Bottom row: Plots of node purity for three different multiscale, hierarchical methods of clustering: average linkage clustering (ALC), single linkage clustering (SLC) and learning by unsupervised nonlinear diffusion (LUND). As the number of leaves/clusters increases, purity is non-decreasing. The purity of the LUND clusters converges more rapidly to the optimal value 1, indicating that high accuracy can be gained by correctly labeling a smaller number of clusters, compared to ALC and SLC
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Figure 4.  In (a), data with natural hierarchical structure is exhibited. The four Gaussians have means $ (0, 0), (0, 2), \left(\frac{3}{2}, 0\right), \left(\frac{3}{2}, 2\right) $. While at one level of granularity there are 4 clusters (shown in (b)), at a coarser level of granularity the top 2 and bottom 2 Gaussians form clusters, leading to a clustering with only 2 clusters (shown in (c))
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Figure 5.  The matrix of pairwise diffusion distances for $ \log_{10}(t) = 1.5 $ and $ \log_{10}(t) = 5 $ are shown in (a) and (b), respectively, illustrating the hierarchical cluster structure in the data. This hierarchical structure introduces ambiguities into the estimation of the number of clusters $ K $ in LUND, as shown (c). For small time, $ \hat{K} = 4 $, while for larger time $ \hat{K} = 2 $
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Figure 6.  LAND labelings of the data with four queries under two different scenarios: small diffusion time (top row) and large diffusion time (bottom row), and four latent clusters (first column) and two latent clusters (second column). The clusters are closer in the horizontal direction than in the vertical direction, from whence the hierarchical structure is derived. In all cases, LAND is able to to correctly label the data with just four queries, one for each Gaussian
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Figure 7.  Experimental results on the synthetic datasets introduced in Figure 3. We see that LAND achieves rapid convergence to perfect labeling accuracy, compared to much slower convergence for the two comparison methods
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Figure 8.  The Salinas A dataset consists of $ 83 \times 86 = 7138 $ pixels in $ D = 224 $ dimensions. The image has spatial resolution $ 3.7 $m/pixel, and was recorded over Salinas, USA by the Aviris sensor. The six labelled classes are arranged in diagonal rows, and are quite spatially regular. The sum across all spectral bands is shown in (a), and the labeled ground truth is shown in (b), with pixels having the same class being given the same color
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Figure 9.  Pavia data consists of a $ 270\times 50 = 13500 $ subset of the full Pavia data set. The image has spatial resolution $ 1.3 $m/pixel, and was recorded over Pavia, Italy by the ROSIS sensor. It consists of 6 spatial classes, some of which are quite well-spread out in the image. The sum across all spectral bands is shown in (a), and the labeled ground truth is shown in (b), with pixels having the same class being given the same color
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Figure 10.  The active learning results for the Salinas A and Pavia datasets are shown in (a) and (b), respectively. In both cases, the LAND algorithm strongly outperforms the modified LAND variant using randomly selected training data, and the CBAL algorithm. In particular, LAND is able to achieve a significant improvement in accuracy with a very small number of labels
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