Graphs are a prevalent tool in data science, as they model the inherent structure of the data. They have been used successfully in unsupervised and semi-supervised learning. Typically they are constructed either by connecting nearest samples, or by learning them from data, solving an optimization problem. While graph learning does achieve a better quality, it also comes with a higher computational cost. In particular, the current state-of-the-art model cost is for samples. In this paper, we show how to scale it, obtaining an approximation with leading cost of , with quality that approaches the exact graph learning model. Our algorithm uses known approximate nearest neighbor techniques to reduce the number of variables, and automatically selects the correct parameters of the model, requiring a single intuitive input: the desired edge density.
Large Scale Graph Learning from Smooth Signals
Vassilis Kalofolias111Part of this work is in the thesis of the first author . Part of this work was done while the second author was at LTS2. Nathanaël Perraudin111Part of this work is in the thesis of the first author . Part of this work was done while the second author was at LTS2. Signal Processing Laboratory 2 (LTS2 - EPFL) Ecole Polytechnique Fédérale de Lausanne Station 11, 1015 Lausanne, Switzerland Swiss Data Science Center (SDSC - ETHZ) Eidgenössische Technische Hochschule Zürich Universitätstrasse 25, 8006 Zürich, Switzerland
Graphs are an invaluable tool in data science, as they can capture complex structures inherent in seemingly irregular high-dimensional data. While classical applications of graphs include data embedding, manifold learning, clustering and semi-supervised learning [34, 2, 29], they were later used for regularizing various machine learning models, for example for classification, sparse coding, matrix completion, or PCA [32, 33, 16, 28].
More recently, the ability of graphs to capture the structure of seemingly unstructured data, drew the attention of the deep learning community. While convolutional neural networks (CNNs) were highly successful for learning image representations, it was not obvious how to generalize them for high dimensional irregular domains, were standard convolution is not applicable. Graphs gave the solution to bridge the gap between irregular data and CNNs through the generalization of convolutions on graph structures [6, 17, 22, 20]. While it is clear that the graph quality is important in such applications [10, 6], the question of how to optimally construct a graph remains an open problem.
The quality of the graph arguably plays an important role in the success of such methods. The first applications mostly used weighted -nearest neighbors graphs (-NN) [34, 2, 29], but the last few years more sophisticated methods of learning graphs from data were proposed. Today, graph learning, or network inference, has become an important problem itself [30, 5, 12, 19, 11, 7, 15].
Unfortunately, graph learning is computationally too costly for large-scale applications that often need graphs between millions of samples. The current state of the art learning models for weighted undirected graphs [7, 15] cost per iteration for nodes, while previous solutions are even more expensive. Furthermore, they come with parameters that control sparsity, and choosing the correct ones adds an extra burden making them prohibitive for applications with more than a few thousands of nodes.
Large-scale applications can only resort to approximate nearest neighbor methods (A-NN), e.g. [25, 21], that run with a leading cost of . This is remarkably low if we consider that computing a simple -NN graph, or even the pairwise distance matrix between all samples costs . However, the quality of A-NN should be expected to be slightly worse than the quality of -NN, that is already not as good as if we would learn the graph from data.
In this paper, we propose the first scalable graph learning method, with the same leading cost as A-NN, and with quality that approaches state-of-the-art graph learning. Our method leverages A-NN graphs to effectively reduce the number of variables, and the state-of-the-art graph learning model by Kalofolias  in order to achieve the best of both worlds: low cost and good quality. In Figure 1 we illustrate the advantage of our solution compared to the current state-of-the-art. Note that while the standard model costs the same regardless of the graph density , our solution benefits from the desired graph sparsity to reduce computation.
One of our key contributions is to provide a method to automatically select the parameters of the model by Kalofolias  given a desired graph sparsity level. Like in -NN, the user can choose the number of neighbors , without performing grid search over two parameters. Using our scheme, we can learn a 1-million-nodes graph with a desired sparsity level on a desktop in half an hour, with a simple Matlab implementation.
2 Graph Learning from Smooth Signals
A widely used assumption for data residing on graphs is that values change smoothly across adjacent nodes. The smoothness of a set of vectors on a given weighted undirected graph is usually quantified by the Dirichlet energy 
where denotes the weight of the edge between nodes and , is the graph Laplacian, and is the diagonal weighted degree matrix.
Regularization using the Dirichlet energy has been used extensively in machine learning, to enhance for example image processing [9, 33], non-negative matrix factorization [4, 3], matrix completion , or principal component analysis (PCA) [13, 27, 28].
In these methods the Dirichlet energy is minimized w.r.t. matrix given the Laplacian . On the other hand, we can learn a graph under the assumption that is smooth on it, by minimizing the same energy w.r.t. , when is given.
The first works for graph learning focused on learning the weights of a fixed k-nearest neighbor pattern , learning a binary pattern  or the whole adjacency matrix . The idea of minimizing Dirichlet energy for graph learning is more recent [19, 11, 7, 15]. In the last work, Kalofolias  proposed a unified model for learning a graph from smooth signals, that reads as follows:
Here, , denotes the Hadamard product, and the first term is equal to . The optimization is over the set of valid adjacency matrices (non-negative, symmetric, with zero diagonal).
The role of matrix function is to prevent from obtaining a trivial zero value, control sparsity, and impose further structure, depending on the data and the application. Kalofolias obtained state-of-the-art results using
where if condition holds, otherwise. In the sequel we call this the model. Since is the node degrees’ vector, the log model (3) prevents the formation of disconnected nodes due to the logarithmic barrier, while the model (4) controls sparsity by penalizing large degrees due to the first term. The choice therefore depends on the data and application in question.
3 Constrained edge pattern
In traditional graph learning, all possible edges between nodes are considered, that results in a cost of at least computations per iteration. Often, however, we need graphs with a roughly fixed number of edges per node, like in -NN graphs. It is natural to then ask ourselves whether the cost of graph learning can be reduced, reflecting the final desired graph sparsity.
where we optimize in the constrained set of adjacency matrices . After reducing the set of edges to , it suffices to solve the modified problem (5) Following Kalofolias , we can rewrite the problem as
where is the Lipschitz constant of . Note that we gather all free parameters of the adjacency matrix in a vector of size only , that is, the number of allowed edges, each counted only once. Accordingly, in we only keep the corresponding pairwise distances from matrix . The linear operator is also modified, keeping only the columns corresponding to the edges in .
In this form, the problem can be solved by the primal dual techniques by Komodakis and Pesquet . The cost of the dual step, operating on the dual variable (degrees vector) remains . However, the cost of the primal step, as well as the cost of applying the modified operator in order to exchange between the primal and dual spaces is instead of of the initial algorithm 1 by Kalofolias , reducing the overall complexity.
In some cases, a pattern of allowed edges can be induced by constraints of the model, for example sensor networks only assume connections between geographically nearby sensors. In most applications, however, a constrained set is not known beforehand, and we need to approximate the edge support of the final learned graph in order to reduce the number of variables. To this end, we propose using approximate nearest neighbors graphs to obtain a good approximation. While computing a -NN graph needs computations, approximate nearest neighbors (A-NN) algorithms [23, 24, 25, 21] offer a good compromise between accuracy and speed. Specifically, A-NN methods scale gracefully with the number of nodes , the fastest ones having an overall complexity of for -dimensional data.
When approximating the support of the final edges of a graph, we prefer to have false positives than false negatives. We thus start with an initial support with a larger cardinality than that of the desired final graph, and let the weight learning step automatically select which edges to set to zero. Precisely, we select a set with cardinality , where is the desired number of neighbors per node and a small multiplicative factor. By setting the sparsity parameters correctly, the graph learning step will only keep the final edges, setting the less important or wrong edges to zero. The bigger the factor , the more freedom the learning algorithm has to select the right edges. If in the end of the graph learning it occurs that many nodes still have all allowed edges set to positive values, it is an indication that we should have given a more generous set of allowed edges.
3.1 Overall complexity
Asymptotically, the cost of learning a -A-NN graph is for a graph of nodes and data with dimensionality , while additionally learning the edge weights costs per iteration. The overall complexity is therefore for iterations. For large , the dominating cost is asymptotically the one of computing the A-NN and not the cost of learning the weights on the reduced set.
4 Automatic parameter selection
A major problem of models (3) and (4) is the choice of meaningful parameters , as performing grid search increases significantly the computational complexity. In this section we show how this burden can be completely avoided for model (3). We do this in two steps. First, we show that the sparsity depends effectively on a single parameter, and second, we propose a method to set this parameter from the desired connectivity. Our method is based on predicting exactly the number of edges of any given node for a given parameter value, if we relax the symmetricity constraint of .
4.1 Reduction to a single optimization parameter
Let denote the solution of model (3) for input distances and parameters . Then the same solution can be obtained with fixed parameters and , by multiplying the input distances by and the resulting edges by :
Apply [15, Prop. 2], with and divide all operands by the same constant .
Proposition 1 shows that the two parameter spaces and are equivalent. While the first one is convenient to define (3), the second one makes the sparsity analysis and the application of the model simpler. In words, all graphs that can be learned by model (3) can be equivalently computed by multiplying the initial distances by , using them to learn a graph with fixed parameters , and multiplying all resulting edges by the same constant .
This property allows us to tune a single parameter that controls the sparsity of the solution instead of and . The larger is, the greater the pairwise distances between nodes and therefore the sparser the edges, as connections between distant nodes are penalized. This result is also important for graph-based applications that are multiplicative scale invariant. In the latter, multiplying all edges by the same constant does not change the functionality of the graph. In other cases, we want to explicitly normalize the graph to a specific size, for example setting as in , or making sure that . In these cases, the user needs only search for the value of that will obtain the desired sparsity and then multiply the graph with the appropriate constant.
4.2 Setting the remaining regularization parameter
The last step for automatizing parameter selection is to find a relation between and the desired sparsity (the average number of neighbors per node). We first analyze the sparsity level with respect to for each node independently. Once the independent problems are well characterized, we propose an empirical solution to obtain a global value of providing approximately the desired sparsity level.
4.2.1 Sparsity analysis for one node
In order to analyze the sparsity of the graphs obtained by the model (3), we take one step back and drop the symmetricity constraint. The problem becomes separable and we can focus on only one node. Keeping only one column of matrix , we arrive to the simpler optimization problem
The above problem also has only one parameter that controls sparsity, so that larger values of yield sparser solutions . Furthermore, it enjoys an analytic solution if we sort the elements of , as we prove with the next theorem.
Suppose that the input vector is sorted in ascending order. Then the solution of problem (8) has the form
The set corresponds to the indices of the smallest distances and is the cumulative sum of the smallest distances in , .
We provide the proof of Theorem 1 after presenting certain intermediate results. In order to solve Problem (8) we first introduce a slack variable for the inequality constraint, so that the KKT optimality conditions are
the optimum of can be revealed by introducing the term and rewrite (10) as
Then, we split the elements of in two sets, and according to the activity of the inequality constraint (11), so that (inactive) and (active). Note that at the minimum, the elements of will also be sorted in a descending order so that , according to Theorem 1. We first need a condition for an element of to be positive, as expressed in the following lemma:
An element of the solution of problem (8) is in the active set if and only if it corresponds to an element of for which .
We are now ready to proceed to the proof of Theorem 1.
Proof 3 (Theorem 1).
As elements of are sorted in an ascending order, the elements of will be in a descending order. Furthermore, we know from Lemma 1 that all positive will correspond to . Then, supposing that we have the following ordering:
In words, the vector will have sorted elements so that the first are positive and the rest are non-positive. Furthermore, we know that the elements of in the optimal have to be for all inactive variables , therefore . The remaining elements of will be by definition of the active set:
What remains is to find an expression to compute for any given . Keeping ordered in ascending order, let the cumulative sum of be Then, from the definition of and using the structure of we have
which has only one positive solution,
4.2.2 Parameter selection for the non-symmetric case
While Theorem 1 gives the form of the solution for a known , the latter cannot be known a priori, as it is also a function of . For this, we propose Algorithm 1 that solves this problem, simultaneously finding and in iterations. This algorithm will be needed for automatically setting the parameters for the symmetric case of graph learning.
As is the number of non-zero edges per node, it can be assumed to be a small number, like the ones used for nearest neighbor graphs. Therefore, it is cheap to incrementally try all values of starting from until we find the correct one, as Algorithm 1 does. Once we try a value of that satisfies , all KKT conditions hold and we have found the solution to our problem. A similar algorithm has been proposed in  for projecting a vector on the probability simplex, that could be used for a similar analysis for the -degree constraints model (4).
Most interestingly, using the form of the solution given by Theorem 1 we can solve the reverse problem: If we know the distances vector and we want a solution with exactly non-zero elements, what should the parameter be? The following theorem answers this question, giving intervals for as a function of , and its cumulative sum .
Let , the result of problem (8) has exactly non-zero elements.
See Appendix A.
The idea of Theorem 2 is illustrated in the left part of Figure 2. For this figure we have used the distances between one image of MNIST and 999 other images. For any given sparsity level we can know what are the intervals of the valid values of just by looking at the pairwise distances.
4.2.3 Parameter selection for the symmetric case
In order to approximate the parameter that gives the desired sparsity of , we use the above analysis for each row or column separately, omitting the symmetricity constraint. Then, using the arithmetic mean of the bounds of we obtain a good approximation of the behaviour of the full symmetric problem. In other words, to obtain a graph with approximately edges per node, we propose to use the following intervals:
where is obtained by sorting each column of in increasing order, and . The above expression is the arithmetic mean over all minimum and maximum values of that would give a -sparse result if we were to solve problem (8) for each of columns separately, according to Theorem 2. Even though the above approach does not take into account the symmetricity constraints, it gives surprisingly good results in the vast majority of the cases.
In our experiments we wish to answer questions regarding (1) the approximation quality of our large scale model, (2) the quality of our automatic parameter selection, (3) the benefit from learning versus A-NN for large scale applications and (4) the scalability of the model. We perform experiments using 4 real datasets, namely MNIST111We use the 60000 images of the MNIST training dataset., USPS, US Census 1990222The dataset is available at the UCI machine learning repository and consists of approximately 2.5 million samples of 68 features. https://archive.ics.uci.edu/ml/datasets/US+Census+Data+(1990), and Google’s word2vec word representation333The dataset consists of the most used words in English (https://research.googleblog.com/2006/08/all-our-n-gram-are-belong-to-you.html). It uses the Google word2vec features (https://code.google.com/archive/p/word2vec/).. Further datasets and experiments can be found in the Appendix. All timing results reported are computed using a standard desktop computer.
5.1 Effectiveness of automatic parameter selection
The plot in the middle of Figure 2 shows the approximate bounds obtained by eq. (4.2.3) for 1000 images of the MNIST dataset, in comparison to the actual sparsity obtained for each choice of (red line). We repeat the same experiment for images from the USPS dataset (right plot of Figure 2), and for more datasets in the Appendix B.2. Please note, that in the rare cases that the actual sparsity is outside the predicted bounds, we already have a good starting point for finding a good . Note also that small fluctuations in the density are tolerated, for example in -NN graphs we always obtain results with slightly more than edges due to the fact that is symmetric.
5.2 Graph between images of MNIST
To assess the quality of our model, we first learn the graph between the all images of the training set of MNIST. Using the image labels, we first show that, for the same average degree, the amount of wrong edges (connecting wrong classes) is much lower in learned graphs than in A-NN graphs. Second, we show that graph learning improves the performance of semi-supervised learning.
5.2.1 Edge quality
For graph learning, MNIST is an interesting dataset because it has a relatively uniform sampling between numbers of different labels, except for the digits “1” that are more densely sampled. In other words, the intra-class distances between the digits “1” is in average much smaller than for other digits (see also Appendix B.1). This affects the results of graph learning as we see in the sequel.
We analyze the quality of the connections retrieved by the large-scale log and models and we compare with an A-NN graph as a baseline. Each graph is normalized so as to have equal total “connectivity” . In Figure 3, we plot the histograms of the connectivity ratio that is spent for edges between images of each label. The last bar corresponds to the total ratio of connectivity wasted on wrong edges (edges between any pair of images with different labels).
Given the almost uniform distribution of the labels and distances in the MNIST dataset, we expect to see a similar behavior in the connectivity between the images, with a small variation due to different average distances (with label “1” more connected). Ideally, the wrong edges should be minimal.
The model (4) generally fails to give consistent connectivities across different labels, being too sensitive to the average distances and assigns the vast majority of its connectivity only to the label “1” that has the smallest intra-label image distance. This effect becomes smaller when more dense graphs (30 edges per node, yellow bars) are sought. On the other hand, the log model does not suffer from this problem and gives consistent connectivities without depending too much on the sampling density. Similarly for a -NN graph, the connectivity is constant across all labels. Note, however, that we use it here only as a baseline, and that a weighted -NN graph would also follow the intra-label distances depending on its weighting scheme.
Figure 4 summarizes the number of wrong edges with respect to . Clearly, learned graphs have much less wrong edges at any given graph density. The model has the least number of wrong edges. However, as shown in Figure 3, the majority of its edges connects the digit one and digits such as are almost not connected. On the contrary, the log model is much less affected from this drawback, while keeping a relatively low amount of wrong edges.
5.2.2 Semi-supervised learning
Using the graph learned in the previous subsection, we perform a semi-supervised learning experiment on the MNIST dataset. We use only 1 of the labels as observations, and apply label propagation  to predict the rest of the labels. The results are plotted in Figure 5. Again, the log model performs best. To have a fair comparison we used the best weighting scheme for -NN, while the quality of the label propagation did not vary significantly by changing weighting schemes.
Given the performance of the A-NN graph, one might wonder why pay the additional cost of learning a graph only for a small improvement in classification. Note, however, that the additional cost is not significant. Asymptotically, the cost of learning an A-NN graph is for a graph of nodes and data with dimensionality , while additionally learning the edge weights costs only . The asymptotical complexity is thus dominated by the cost of computing an A-NN, and not by the cost of learning the weights. For the relatively small size of the problem we solve, this corresponds to seconds for the A-NN graph (using compiled C code), and additional seconds for our graph learning (using a Matlab-only implementation)444Our implementation is available in the GSPBox . A tutorial based on this paper can be found in https://epfl-lts2.github.io/gspbox-html/doc/demos/gsp_demo_learn_graph_large.html.. With a full C implementation for our algorithm, we would expect the second number to be significantly smaller. Furthermore, for really large scale problems the asymptotical complexity would show that the bottleneck is not the graph learning.
5.3 Graph between words using word2vec representations
Motivated by the application of graphs between words in graph convolutional neural networks , we investigate the quality of our large scale graph learning in this type of data. For this, we learn a graph connecting similar words using Google word2vec features. A first observation is made in Figure 7, where we plot the diameters of different graphs for different density levels. We see that given the same density, the learned graph has a significantly larger diameter than both -NN and A-NN. This indicates that our learned graph is closer to a manifold-like structure, unlike the other two types that are closer to a small-world graph. Manifold-like graphs reveal better the structure of data, while in small world graphs are related to randomness .
We see this also qualitatively in Figure 6, where we plot nodes within two hops from the word "use" in the three types of graphs. We see that the NN graphs span a larger part of the entire graph just with hops, the A-NN being closer to small-world. While the -NN graph does better in terms of quality, it seems to do worse than our graph learning, that is actually cheaper to compute. In this sense, graph learning is a response to the randomness of A-NN while improving in speed upon -NN. Additionally, graph learning seems to assign more meaningful weights as we show in Table 1 of the Appendix B.3. Note that we do not include results for the model as it performs very poorly, leaving many disconnected nodes.
5.4 Approximation quality of large scale model
When computing , we use an approximate nearest neighbor (A-NN) graph using the publicly available FLANN library555Compiled C code run through Matlab, available from http://www.cs.ubc.ca/research/flann/. that implements the work of Muja and Lowe . To learn a graph with on average neighbors per node (-NN), we first compute a -A-NN graph and use its edges as . The graph is then learned on this subset. The choice of the number of allowed edges does not only affect the time needed to learn the graph, but also its quality. A too restrictive choice might prevent the final graph form learning useful edges.
In Figure 8, we study the effect of this restriction on the final result. The vertical axis is the relative error between our approximate log model and the actual log model by Kalofolias  when learning a graph between 1000 images of MNIST, averaged over 10 runs. Note that the result will depend on the A-NN algorithm used, while a comparison between different types of A-NN is beyond the scope of this paper.
5.5 Computation time
To show the scalability of our algorithm, we plot the time needed for learning different graph sizes between words. As we see in Figure 1, the cost is almost linear for our method, but quadratic for the original work of Kalofolias . We also tested our Matlab implementation on K and M samples of the US census dataset. Setting , it used minutes to perform iterations of graph learning. In Figure 11 of Appendix B.4, we also illustrate the linear scalability of our model w.r.t. the size of the set of allowed edges.
We propose the first scalable solution to learn a weighted undirected graph from data, based on A-NN and the current state-of-the-art graph learning model. While it costs roughly as mush as A-NN, it achieves quality very close to state-of-the-art. Its success is based primarily on reducing the variables used for learning and secondarily on selecting all parameters controlling the graph sparsity completely automatically. We assess its quality and scalability by providing an extensive set of experiments on many real datasets. Learning a graph of 1 million nodes only takes 29 minutes using our simple Matlab implementation on a desktop computer.
A MATLAB implementation of our large scale graph learning is available in the GSPBOX . A tutorial with most important functionality is available here: https://epfl-lts2.github.io/gspbox-html/doc/demos/gsp_demo_learn_graph_large.html
The authors would like to especially thank Pierre Vandergheynst for his helpful comments during the preparation of this work at LTS2.
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Appendix A Proof of Theorem 2
From the proof of Theorem 1, we know that if and only if . We can rewrite this condition as
As is constrained to be positive, the only values that satisfy the above inequalities are the ones proposed in the theorem.
Appendix B Experiments
b.1 MNIST irregular intra-class distances
Figure 9 illustrates one irregularity of the MNIST dataset. One could expect that the average distance between two digits of the same class (intra-class) is more or less independent of the class. Nevertheless, for the MNIST dataset, the distance between the is significantly smaller than the one of the other digits. For this reason, the L2 model connects significantly more the digits than the others.
b.2 Accuracy of the tuning of the parameter
We already saw in Figure 2 (middle) that for the MNIST dataset eq. (4.2.3) predicts very well the sparsity of the final graph for any choice of . This is further illustrated on the USPS and ATT faces datasets in Figure 10. Note, that in the rare cases that the actual sparsity is outside the predicted bounds, we already have a good starting point for finding a good . For example, in the COIL dataset, if we want a graph with edges per node we will set , obtaining instead a graph with edges per node. This kind of fluctuations are usually tolerated, while even in -NN we always obtain graphs with more than edges due to the fact that is symmetric.
b.3 Connectivity example of the graph of words
In Table 1, we look in more detail at the graph constructed from the word2vec features. We present the connectivity for the word "glucose" and "academy". Looking at different words, we observe that the learned graph is able to associate meaningful edge weights to the different words according to the confidence of their similarity.
b.4 MNIST Computational time
The cost of learning a graph with a subset of allowed edges is linear to the size of the set as illustrated in Figure 11. For this experiment, we use the MNIST data set. To learn a graph with approximately edges per node, we needed seconds to compute , and seconds to learn the final graph of nodes (around 250 iterations). Note that the time necessary to search for the nearest neighbors is in the same order of magnitude than the learning process.
b.5 Another measure to test the parameter
The percentage of nodes of the final graph that have the maximum number of edges allowed by is a measure of this quality deterioration due to the edge restriction imposed by the A-NN. Using the MNIST dataset, it is plotted for our large scale log-degree model in the bottom part of Figure 12. In the horizontal axis we have , and different lines correspond to different densities of the A-NN graph, from to . We find that an A-NN of (i.e. ) is a good compromise between time and quality, and use it for most of the experiments of this paper.