In a number of modern applications, the data appear to cluster near some low-dimensional structures. In the particular setting of manifold learning [51, 47, 7, 22, 17], the data are assumed to lie near manifolds embedded in Euclidean space. When multiple manifolds are present, the foremost task is separating them, meaning the recovery of the different components of the data associated with the different manifolds. Manifold clustering naturally occurs in the human visual cortex, which excels at grouping points into clusters of various shapes [41, 21]. It is also relevant for a number of modern applications. For example, in cosmology, galaxies seem to cluster forming various geometric structures such as one-dimensional filaments and two-dimensional walls [52, 39]
. In motion segmentation, feature vectors extracted from moving objects and tracked along different views cluster along affine or algebraic surfaces[35, 23, 53, 10]
. In face recognition, images of faces in fixed pose under varying illumination conditions cluster near low-dimensional affine subspaces[31, 6, 19], or along low-dimensional manifolds when introducing additional poses and camera views.
In the last few years several algorithms for multi-manifold clustering were introduced; we discuss them individually in Section 1.3.3. We focus here on spectral clustering methods, and in particular, study a prototypical multiway method relying on local linear approximations, with precursors appearing in [12, 1, 48, 2, 27]. We refer to this method as Higher-Order Spectral Clustering (HOSC). We establish theoretical guarantees for this method within a standard mathematical framework for multi-manifold clustering. Compared with all other algorithms we are aware of, HOSC is able to separate clusters that are much closer together; equivalently, HOSC is accurate under much lower sampling rate than any other algorithm we know of. Roughly speaking, a typical algorithm for multi-manifold clustering relies on local characteristics of the point cloud in a way that presupposes that all points, or at least the vast majority of the points, in a (small enough) neighborhood are from a single cluster, except in places like intersections of clusters. In contrast, though HOSC is also a local method, it can work with neighborhoods where two or more clusters coexist.
1.1 Higher-Order Spectral Clustering (HOSC)
We introduce our higher-order spectral clustering algorithm in this section, tracing its origins to the spectral clustering algorithm of Ng et al.  and the spectral curvature clustering of Chen and Lerman [12, 11].
Spectral methods are based on building a neighborhood graph on the data points and partitioning the graph using its Laplacian [22, 34], which is closely related to the extraction of connected components. The version introduced by Ng et al.  is an emblematic example—we refer to this approach as SC. It uses an affinity based on pairwise distances. Given a scale parameter and a kernel , define
( denotes the Euclidean norm.) Standard choices include the heat kernel and the simple kernel . Let denote the data points. SC starts by computing all pairwise affinities , with , for . It then computes the matrix , where is the degree of the th point in the graph with similarity matrix . Note that is the corresponding normalized Laplacian. Providing the algorithm with the number of clusters , SC continues by extracting the top eigenvectors of , obtaining a matrix , and after normalizing its rows, uses them to embed the data into . The algorithm concludes by applying -means to the embedded points. See Algorithm 1 for a summary.
|: the data points|
|: the affinity scale|
|: the number of clusters|
|A partition of the data into disjoint clusters|
Compute the affinity matrix, with .
|2: Compute the , where .|
|3: Extract , the top eigenvectors of .|
|4: Renormalize each row of to have unit norm, obtaining a matrix .|
|5: Apply -means to the row vectors of in to find clusters.|
|6: Accordingly group the original points into disjoint clusters.|
Spectral methods utilizing multiway affinities were proposed to better exploit additional structure present in the data. The spectral curvature clustering (SCC) algorithm of Chen and Lerman [12, 11] was designed for the case of hybrid linear modeling where the manifolds are assumed to be affine, a setting that arises in motion segmentation . Assuming that the subspaces are all of dimension —a parameter of the algorithm, SCC starts by computing the (polar) curvature of all -tuples, creating an
-tensor. The tensor is then flattened into a matrixwhose product with its transpose, , is used as an affinity matrix for the spectral algorithm SC. (In practice, the algorithm is randomized for computational tractability.) Kernel spectral curvature clustering (KSCC)  is a kernel version of SCC designed for the case of algebraic surfaces.
The SCC algorithm (and therefore KSCC) is not localized in space as it fits a parametric model that is global in nature. The method we study here may be seen as a localization of SCC, which is appropriate in our nonparametric setting since the manifolds resemble affine surfaces locally. This type of approach is mentioned in publications on affinity tensors[1, 48, 2, 27] and is studied here for the first time, to our knowledge. As discussed in Section 4, all reasonable variants have similar theoretical properties, so that we choose one of the simplest versions to ease the exposition. Concretely, we consider a multiway affinity that combines pairwise distances between nearest neighbors and the residual from the best -dimensional local linear approximation. Formally, given a set of points, , define
where for a subset and denotes the set of -dimensional affine subspaces in . In other words, is the width of the thinnest tube (or band) around a -dimensional affine subspace that contains . (In our implementation, we use the mean-square error; see Section 3.) Given scale parameters and a kernel function , define the following affinity: if are not distinct; otherwise:
where is the diameter of . See Figure 1 for an illustration.
Given data points and approximation dimension , we compute all -way affinities, and then obtain pairwise similarities by clique expansion  (note that several other options are possible [12, 48, 27]):
Though it is tempting to choose equal to , a larger allows for more tolerance to weak separation and small sampling rate. The down side is what appears to be an impractical computational burden, since the mere computation of in (4) requires order flops. In Section 1.4, we discuss how to reduce the computational complexity to flops, essentially without compromising performance.
Once the affinity matrix is computed, the SC algorithm is applied. We call the resulting procedure higher-order spectral clustering (HOSC), summarized in Algorithm 2. Note that HOSC is (essentially) equivalent to SC when , and equivalent to SCC when .
|: the data points|
|: the approximation dimension and affinity order|
|: the affinity scales|
|: the number of clusters|
|A partition of the data into disjoint clusters|
|1: Compute the affinity matrix according to (4).|
|2: Apply SC (Algorithm 1).|
1.2 Generative Model
It is time to introduce our framework. We assume a generative model where the clusters are the result of sampling points near surfaces embedded in an ambient Euclidean space, specifically, the -dimensional unit hypercube . For a surface and , define its -neighborhood as
The reach of is the supremum over such that, for each , there is a unique point realizing . It is well-known that, for submanifolds, the reach bounds the radius of curvature from below [20, Lem. 4.17]. For a connection to computational geometry, the reach coincides with the condition number introduced in  for submanifolds without boundary. Let denote the -dimensional Hausdorff measure, and the boundary of within . For an integer and a constant , let be the class of -dimensional, connected, submanifolds of and , and if has a boundary, is a -dimensional submanifold with . Given surfaces and , we generate clusters by sampling points uniformly at random in , the -neighborhood of in , for all . We call the jitter level. Except for Section 2.3, where we allow for intersections, we assume that the surfaces are separated by a distance of at least , i.e.
In that case, by the triangle inequality, the actual clusters are separated by at least , i.e.
We assume that the clusters are comparable in size by requiring that for all , for some finite constant . Let denote the data points thus generated. See Figure 2 for an illustration.
Given data , we aim at recovering the clusters . Formally, a clustering algorithm is a function taking data , and possibly other tuning parameters, and outputs a partition of . We say that it is ‘perfectly accurate’ if the output partition coincides with the original partition of into . Our main focus is on relating the sample size and the separation requirement in (5) (in order for HOSC to cluster correctly), and in particular we let and vary with . This dependency is left implicit. In contrast, we assume that are fixed. Also, we assume that are known throughout the paper (except for Section 2.1 where we consider their estimation). Though our setting is already quite general, we discuss some important extensions in Section 4.
We will also consider the situation where outliers may be present in the data. By outliers we mean points that were not sampled near any of the underlying surfaces. We consider a simple model where outliers are points sampled uniformly in for some , in general different from . That is, outliers are at least a distance away from the surfaces. We let denote the number of outliers, while still denotes the total number of data points, including outliers. See Figure 3 for an illustration.
1.3 Performance in terms of Separation and Robustness
1.3.1 Performance of SC
A number of papers analyze SC under generative models similar to ours [3, 54, 44, 40], and the closely related method of extracting connected components of the neighborhood graph [3, 37, 9, 36]. The latter necessitates a compactly supported kernel and may be implemented via a union-of-balls estimator for the support of the density . Under the weaker (essentially Lipschitz) regularity assumption
Arias-Castro  shows that SC with a compactly supported kernel is accurate if
( denotes the maximum of and and if as ). With the heat kernel, the same result holds up to a multiplicative factor. See also [37, 36], which prove a similar result for the method of extracting connected components under stronger regularity assumptions. At the very least, (7) is necessary for the union-of-balls approach and for SC with a compactly supported kernel, because is the order of magnitude of the largest distance between a point and its closest neighbor from the same cluster . Note that (6) is very natural in the context of clustering as it prevents from being too narrow in some places and possibly confused with two or more disconnected surfaces. And, when in (6) is large enough and is small enough, it is satisfied by any surface belonging to . Indeed, such a surface resembles an affine subspace locally and (6) is obviously satisfied for an affine surface.
When outliers may be present in the data, as a preprocessing step, we identify as outliers data points with low connectivity in the graph with affinity matrix , and remove these points from the data before proceeding with clustering. (This is done between Steps 1 and 2 in Algorithm 1.) In the context of spectral clustering, this is very natural; see, e.g., [12, 37, 3]. Using the pairwise affinity (1), outliers are properly identified if satisfies the lower bound in (7) and if the sampling is dense enough, specifically ,
When the surfaces are only required to be of Lipschitz regularity as in (6), we are not aware of any method that can even detect the presence of clusters among outliers if the sampling is substantially sparser.
1.3.2 Performance of HOSC
Methods using higher-order affinities are obviously more complex than methods based solely on pairwise affinities. Indeed, HOSC depends on more parameters and is computationally more demanding than SC. One, therefore, wonders whether this higher level of complexity is justified. We show that HOSC does improve on SC in terms of clustering performance, both in terms of required separation between clusters and in terms of robustness to outliers.
Our main contribution in this paper is to establish a separation requirement for HOSC which is substantially weaker than (7) when the jitter is small enough. Specifically, HOSC operates under the separation
where denotes the minimum of and , and is the separation required for SC with a compactly supported kernel, defined in (7). This is proved in Theorem 1 of Section 2. In particular, in the jitterless case (i.e. ), the magnitude of the separation required for HOSC is (roughly) the square of that for SC at the same sample size; equivalently, at a given separation, HOSC requires (roughly) the square root of the sample size needed by SC to correctly identify the clusters.
That HOSC requires less separation than SC is also observed numerically. In Figure 4 we compare the outputs of SC and HOSC on the emblematic example of concentric circles given in  (here with three circles). While the former fails completely, the latter is perfectly accurate. Indeed, SC requires that the majority of points in an -ball around a given data point come from the cluster containing that point. In contrast, HOSC is able to properly operate in situations where the separation between clusters is so small, or the sampling rate is so low, that any such neighborhood is empty of data points except for the one point at the center. To further illustrate this point, consider the simplest possible setting consisting of two parallel line segments in dimension , separated by a distance , specifically, and . Suppose points are sampled uniformly on each of these line segments. It is well-known that the typical distance between a point on and its nearest neighbor on is of order ; see . Hence, a method computing local statistics requires neighborhoods of radius at least of order , for otherwise some neighborhoods are empty. From (9), HOSC is perfectly accurate when , say. When the separation is that small, typical ball of radius of order around a data point contains about as many points from as from (thus SC cannot work). See Figure 5 for an illustration.
To make HOSC robust to outliers, we do exactly as described above, identifying outliers as data points with low connectivity in the graph with affinity matrix , this time computed using the multiway affinity (3). The separation and sampling requirements are substantially weaker than (8), specifically, is required to satisfy the lower bound in (9) and the sampling
This is established in Proposition 5, and again, we are not aware of any method for detection that is reliable when the sampling is substantially sparser. For example, when and we are clustering curves () in the plane () (with background outliers), the sampling requirement in (8) is roughly , compared to in (10). In Figure 6
below we compare both SC and HOSC on outliers detection, using the data in Figure4 but further corrupted with outliers.
1.3.3 Other Methods
We focus on comparing HOSC and SC to make a strong point that higher-order methods may be preferred to simple pairwise methods when the underlying clusters are smooth and the jitter level is small. In fact, we believe that no method suggested in the literature is able to compete with HOSC in terms of separation requirements. We quickly argue why.
The algorithm of Kushnir et al. 
is multiscale in nature and is rather complex, incorporating local information (density, dimension and principal directions) within a soft spectral clustering approach. In the context of semi-supervised learning, Goldberg et al.
introduce a spectral clustering method based on a local principal components analysis (PCA) to utilize the unlabeled points. Both methods rely on local PCA to estimate the local geometry of the data and they both operate by coarsening the data, eventually applying spectral clustering to a small subset of points acting as representative hubs for other points in their neighborhoods. They both implicitly require that, for the most part, the vast majority of data points in each neighborhood where the statistics are computed come from a single cluster. Souvenir and Pless suggest an algorithm that starts with ISOMAP and then alternates in EM-fashion between the cluster assignment and the computation of the distances between points and clusters—this is done in a lower dimensional Euclidean space using an MDS embedding. Though this iterative method appears very challenging to be analyzed, it relies on pairwise distances computed as a preprocessing step to derive the geodesic distances, which implicitly assumes that the points in small enough neighborhoods are from the same manifold. Thus, like the SC algorithm, all these methods effectively rely on neighborhoods where only one cluster dominates. This is strong evidence that their separation requirements are at best similar to that of SC. The methods of Haro et al.  and Gionis et al.  are solely based on the local dimension and density, and are powerless when the underlying manifolds are of same dimension and sampled more or less uniformly, which is the focus of this paper. The method of Guo et al.  relies on minimizing an energy that, just as HOSC, incorporates the diameter and local curvature of -tuples, with for curves and for surfaces in 3D, and the minimization is combinatorial over the cluster assignment. In principle, this method could be analyzed with the arguments we deploy here. That said, it seems computationally intractable.
1.4 Computational Considerations
Thus it appears that HOSC is superior to SC and other methods in terms of separation between clusters and robustness to outliers, when the clusters are smooth and the jitter is small. But is HOSC even computationally tractable?
Assume and are fixed. The algorithm starts with building the neighborhood graph (i.e., computing the matrix ). This may be done by brute force in flops. Clearly, this first step is prohibitive, in particular since we recommend using a (moderately) large . However, we may restrict computations to points within distance , which essentially corresponds to using a compactly supported kernel . Hence, we could apply a range search algorithm to reduce computations. Alternatively, at each point we may restrict computations to its nearest neighbors, with , or in a slightly different fashion, adapt the local scaling method proposed in  by replacing in by , where denotes the distance between and its th nearest neighbor. The reason is that the central condition (12) effectively requires that the degree at each point be of order (roughly), which is guaranteed if the -nearest neighbors are included in the computations; see [3, 36] for rigorous arguments leading to that conclusion. In low dimensions, , a range search and -nearest-neighbor search may be computed effectively with kd-trees in flops. In higher dimensions, it is essential to use methods that adapt to the intrinsic dimensionality of the data. Assuming that is small, the method suggested in  has a similar computational complexity. Hence, the (approximate) affinity matrix can be computed in order ; assuming , this is of order . This is within the possible choices for in Theorem 1.
Assume we use the -nearest-neighbor approximation to the neighborhood graph, with . Then computing may be done in flops, since the affinity matrix has at most non-zero coefficients per row. Then extracting the leading eigenvectors of may be done in flops, using Lanczos-type algorithms . Thus we may run the -nearest neighbor version of HOSC in flops, and it may be shown to perform comparably.
The rest of the paper is organized as follows. The main theoretical results are in Section 2 where we provide theoretical guarantees for HOSC, including in contexts where outliers are present or the underlying clusters intersect. We emphasize that HOSC is only able to separate intersecting clusters under very stringent assumptions. In the same section we also address the issue of estimating the parameters that need to be provided to HOSC. In theory at least, they may be chosen automatically. In Section 3 we implemented our own version of HOSC and report on some numerical experiments involving both simulated and real data. Section 4 discusses a number of important extensions, such as when the surfaces self-intersect or have boundaries, which are excluded from the main discussion for simplicity. We also discuss the case of manifolds of different intrinsic dimensions, suggesting an approach that runs HOSC multiple times with different . And we describe a kernel version of HOSC that could take advantage of higher degrees of smoothness. Other extensions are also mentioned, including the use of different kernels. The proofs are postponed to the Appendix.
2 Theoretical Guarantees
Our main result provides conditions under which HOSC is perfectly accurate with probability tending to one in the framework introduced in Section1.2. Throughout the paper, we state and prove our results when the surfaces have no boundary and for the simple kernel , for convenience and ease of exposition. We discuss the case of surfaces with boundaries in Section 4.2 and the use of other kernels in Section 4.5.
We further comment on the theorem. First, the result holds if and is sufficiently large. We state and prove the result when as a matter of convenience. Also, by (11) and (14), the weakest separation requirement is achieved when is at least of order slightly less than so that is of order . However, as discussed in Section 1.4, the algorithm is not computationally tractable unless . This is another reason why we focus on the case where . Regarding the constraints (12)-(13) on and , they are there to guarantee that, with probability tending to one, each cluster is ‘strongly’ connected in the neighborhood graph. Note that the bound on is essentially the same as that required by the pairwise spectral method SC [3, 36]. In turn, once each cluster is ‘strongly’ connected in the graph, clusters are assumed to be separated enough that they are ‘weakly’ connected in the graph. The lower bound (14) quantifies the required separation for that to happen. Note that it is specific to the simple kernel. For example, the heat kernel would require a multiplicative factor proportional to .
So how does HOSC compare with SC? When the jitter is large enough that , we have and the local linear approximation contribution to (3) does not come into play. In that case, the two algorithms will output the same clustering (see Figure 7 for an example).
When the jitter is small enough that , HOSC requires less separation, as demonstrated in Figure 5. Intuitively, in this regime the clusters are sampled densely enough relative to the thickness that the smoothness of the underlying surfaces comes into focus and each cluster, as a point cloud, becomes locally well-approximated by a thin band. We provide some numerical experiments in Section 3 showing HOSC outperforming SC in various settings.
Thus, HOSC improves on SC only when the jitter is small. This condition is quite severe, though again, we do not know of any other method that can accurately cluster under the weak separation requirement displayed here, even in the jitterless case. It is possible that some form of scan statistic (i.e., matched filters) may be able to operate under the same separation requirement without needing the jitter to be small, however, we do not know how to compute it in our nonparametric setting—even in the case of hybrid linear modeling where the surfaces are affine, computing the scan statistic appears to be computationally intractable. At any rate, the separation required by HOSC is essentially optimal when is of order or smaller. A quick argument for the case and goes as follows. Consider a line segment of length one and sample points uniformly at random in its -neighborhood, with . The claim is that this neighborhood contains an empty band of thickness of order slightly less than , and therefore cannot be distinguished from two parallel line segments. Indeed, such band of half-width inside that neighborhood is empty of sample points with probability , which converges to 1 if , and when , this is the case if .
In regards to the choice of parameters, the recommended choices depend solely on . These model characteristics are sometimes unavailable and we discuss their estimation in Section 2.1. Afterwards, we discuss issues such as outliers (Section 2.2) and intersection (Section 2.3).
2.1 Parameter Estimation
In this section, we propose some methods to estimate the intrinsic dimension of the data, the jitter and the number of clusters . Though we show that these methods are consistent in our setting, further numerical experiments are needed to determine their potential in practice.
Compared to SC, HOSC requires the specification of three additional parameters. This is no small issue in practice. In theory, however, we recommend choosing and consistent with their true values, and as functions of , and of order slightly less than . The true unknowns are therefore . We provide estimators for and that are consistent, and an estimator for that is accurate enough for our purposes. Specifically, we estimate and using the correlation dimension  and an adaptation of our own design. The number of clusters is estimated via the eigengap of the matrix .
2.1.1 The Intrinsic Dimension and the Jitter Level
A number of methods have been proposed to estimate the intrinsic dimensionality; we refer the reader to  and references therein. The correlation dimension, first introduced in , is perhaps the most relevant in our context, since surfaces may be close together. Define the pairwise correlation function
The authors of  recommend plotting versus and estimating the slope of the linear part. We use a slightly different estimator that allows us to estimate too, if it is not too small. The idea is to regress on and identify a kink in the curve. See Figure 8 for an illustration.
Though several (mostly ad hoc) methods have been proposed for finding kinks, we describe a simple method for which we can prove consistency. Fix , with . Define
Let . If there is such that
then let be the smallest such ; otherwise, let . Define ; and also , if , and the closest integer to , otherwise.
Consider the generative model described in Section 1.2 with . Assume that and, if there are outliers, assume that . Then the following holds with probability at least : if , then ; if , then ; moreover, if , .
In the context of Proposition 1, the only time that is inconsistent is when is of order or larger, in which case ; this makes sense, since the region is in fact -dimensional if is of order 1. Also, is within a factor of if is not much smaller than .
We now extend this method to deal with a smaller . Consider what we just did. The quantity is the total degree of the -neighborhood graph built in SC. Fixing , we now consider the total degree of the -neighborhood graph built in HOSC. Define the multiway correlation function
Similarly, we shall regress on and identify a kink in the curve (Figure 9 displays such a curve).
Using the multiway correlation function, we then propose an estimator as follows. We assume that the method of Proposition 1 returned , for otherwise we know that is accurate. Choose and . Note that this is the only time we require to be larger than . Let . If there is such that
then let be the smallest one; otherwise, let . We then redefine as .
In the context of Proposition 1, assume that . Then redefining as done above, the following holds with probability at least : if , then ; if , then .
Though our method works in theory, it is definitely asymptotic. In practice, we recommend using other approaches for determining the location of the kink and the slope of the linear part of the pairwise correlation function (in log-log scale). Robust regression methods with high break-down points, like least median of squares and least trimmed squares, worked well in several examples. We do not provide details here, as this is fairly standard, but the figures are quite evocative.
2.1.2 The Number of Clusters
HOSC depends on choosing the number of clusters appropriately. A common approach consists in choosing
by inspecting the eigenvalues of. We show that, properly tuned, this method is consistent within our model.
Compute the matrix in HOSC with the same choice of parameters as in Theorem 1, except that knowledge of is not needed. Set the number of clusters equal to the number of eigenvalues of (counting multiplicity) exceeding . Then with probability at least , this method chooses the correct number of clusters.
We implicitly assumed that and are known, or have been estimated as described in the previous section. The proof of Proposition 3 is parallel to that of [3, Prop. 4], this time using the estimate provided in part (A1) of the proof of Theorem 1. Details are omitted.
Figure 10 illustrates a situation where the number of clusters is correctly chosen by inspection of the eigenvalues, more specifically, by counting the number of eigenvalue in the spectrum of (up to numerical error). This success is due to the fact that the clusters are well-separated, and even then, the eigengap is quite small.
We apply this strategy to more data later in Section 3, and show that it can correctly identify the parameter in some cases (see Figure 14). In general we do not expect this method to work well when the data has large noise or intersecting clusters, though we do not know of any other method that works in theory under our very weak separation requirements.
2.2 When Outliers are Present
So far we have only considered the case where the data is devoid of outliers. We now assume that some outliers may be included in the data as described at the end of Section 1.2. As stated there, we label as outlier any data point with low degree in the neighborhood graph, as suggested in [12, 37, 3]. Specifically, we compute as in Step 2 of HOSC, and then label as outliers points with degree below some threshold. Let slower than any power of , e.g., . We propose two thresholds:
Identify as outliers points with degree:
Identify as outliers points with degree:
Taking up the task of identifying outliers, only the separation between outliers and non-outliers is relevant, so that we do not require any separation between the actual clusters. We first analyze the performance of (O1), which requires about the same separation between outliers and non-outliers as HOSC requires between points from different clusters in (14).
We now analyze the performance of (O2), which requires a stronger separation between outliers and non-outliers, but operates under very weak sampling requirements.
If , so that outliers are sampled everywhere but within the -tubular regions of the underlying surfaces, then both (O1) and (O2) may miss some outliers within a short distance from some . Specifically, (O1) (resp. (O2)) may miss outliers within (resp. within ) from some . Using Weyl’s tube formula , we see that there are order (resp. ) such outliers, a small fraction of all outliers.
The sampling requirement (16) is weaker than the corresponding requirement for pairwise methods displayed in (8). In fact, (16) is only slightly stronger than what is required to just detect the presence of a cluster hidden in noise. We briefly explain this point. Instead of clustering, consider the task of detecting the presence of a cluster hidden among a large number of outliers. Formally, we observe the data
, and want to decide between the following two hypotheses: under the null, the points are independent, uniformly distributed in the unit hypercube; under the alternative, there is a surface such that points are sampled from as described in Section 1.2, while the rest of the points, of them, are sampled from the unit hypercube , again uniformly. Assuming that the parameters and are known, it is shown in [5, 4] that the scan statistic is able to separate the null from the alternative if
We are not aware of a method that is able to solve this detection task at a substantially lower sampling rate, and (16) comes within a logarithmic factor from (17). We thus obtain the remarkable result that accurate clustering is possible within a log factor of the best (known) sampling rate that allows for accurate detection in the same setting.
2.3 When Clusters Intersect
We now consider the setting where the underlying surfaces may intersect. The additional conditions we introduce are implicit constraints on the dimension of, and the incidence angle at, the intersections. We suppose there is an integer and a finite constant such that
(The subscript stands for ‘intersection’.) In addition, we assume that for some ,
(18) is slightly stronger than requiring that has finite -dimensional volume. If the surfaces are affine, it is equivalent to the condition (19), on the other hand, is a statement about the minimum angle at which any two surfaces intersect. For example, if the surfaces are affine within distance of their intersection, then (19) is equivalent to their maximum (principal) angle being bounded from below by . See Figure 11 for an illustration.
Illustration of intersecting surfaces. Though the human eye easily distinguishes the two clusters, the clustering task is a lot harder for machine learning algorithms. The main issue is that there are too many data points at the intersection of the two tubular regions. However, in very special cases HOSCis able to separate intersecting clusters (see Figure 12 for such an example).
The most favorable case is when and . Then with our choice of and in Theorem 1, assuming increases slowly, e.g., , we have if , and partial results suggest this cannot be improved substantially. This constraint on the intersection of two surfaces is rather severe. Indeed, a typical intersection between two (smooth) surfaces of same dimension is of dimension , and if so, only curves satisfy this condition. Figure 12 provides a numerical example showing the algorithm successfully separating two intersecting one-dimensional clusters. Thus, even with no jitter and the surfaces intersecting at right angle, HOSC is only able to separate intersecting clusters under exceptional circumstances. Moreover, even when the conditions of Proposition 6 are fulfilled, the probability of success is no longer exponentially small, but is at best of order . That said, SC does not seem able to properly deal with intersections at all (see also Figure 12). It essentially corresponds to taking in HOSC, in which case never tends to zero.
Though the implications of Proposition 6 are rather limited, we do not know of any other clustering method which provably separates intersecting clusters under a similar generative model. This is a first small step towards finding such a method.
3 Software and Numerical Experiments
We include in this section a few experiments where a preliminary implementation of HOSC outperforms SC, to demonstrate that higher-order affinities can bring a significant improvement over pairwise affinities in the context of manifold clustering.
In our implementation of HOSC, we used the heat kernel . Following the discussion in Section 1.4, at each point we restrict the computations to its nearest neighbors so that we practically remove the locality parameter from the affinity function of (3) and obtain
where is the set of the nearest neighbors of . For computational ease, we used
which can be easily computed using the bottom singular values of the points. Note that, since the results we obtained apply, with changed by a factor, at most. (In the paper, the standard choice for is a power of , while is of order at most , so this factor is indeed negligible.) In practice, we always search a subinterval of for the best working (e.g.,), as suggested in . When the given data contains outliers, the optimal choice of is based on the largest gap between the means of the two sets of degrees (associated to the inliers and outliers), normalized by the maximum degree. The code is available online .
3.1 Synthetic Data
We first generate five synthetic data sets in the unit cube ( or ), shown in Figure 13. In this experiment, the actual number of clusters (i.e. ) and dimension of the underlying manifolds (i.e. ) are assumed known to all algorithms. For HOSC, we fix , and use the subinterval as the search interval of . For SC, we considered two ways of tuning the scale parameter : directly, by choosing a value in the interval (SC-NJW); and by the local scaling method of  (SC-LS), with the number of nearest neighbors . The final choices of these parameters were also based on the same criterion as used by HOSC.
Figure 13 exhibits the clusters found by each algorithm when applied to the five data sets, respectively. Observe that HOSC succeeded in a number of difficult situations for SC, e.g., when the sampling is sparse, or when the separation is small at some locations.
We also plot the leading eigenvalues of the matrix obtained by HOSC on each data set; see Figure 14. We see that in data sets 1, 2, 5, the number of eigenvalue 1 coincides with the true number of clusters, while in 3 and 4 there is some discrepancy between the th eigenvalue and the number 1. Though we do not expect the eigengap method to work well in general, Figure 14 shows that it can be useful in some cases.
Figure 15 displays some experiments including outliers. We simply sampled points from the unit square uniformly at random and added them as outliers to the first three data sets in Figure 13, with percentages 33.3%, 60% and 60%, respectively. We applied SC and HOSC assuming knowledge of the proportion of outliers, and labeled points with smallest degrees as outliers. Choosing the threshold automatically remains a challenge; in particular, we did not test the theory.
We observe that HOSC could successfully remove most of the true outliers, leaving out smooth structures in the data; in contrast, SC tended to keep isolated high-density regions, being insensitive to sparse smooth structures. A hundred replications of this experiment (i.e., fixing the clusters and adding randomly generated outliers) show that the True Positive Rates (i.e., percentages of correctly identified outliers) for (SC, HOSC) are (58.1% vs 67.7%), (75.4% vs 86.8%) and (76.8% vs 88.0%), respectively.
3.2 Real Data
We next compare SC and HOSC using the two-view motion data studied in [10, 46]. This data set contains 13 motion sequences: (1) boxes, (2) carsnbus3, (3) deliveryvan, (4) desk, (5) lightbulb, (6) manycars, (7) man-in-office, (8) nrbooks3, (9) office, (10) parking-lot, (11) posters-checkerboard, (12) posters-keyboard, and (13) toys-on-table; and each sequence consists of two image frames of a 3-D dynamic scene taken by a perspective camera (see Figure 16 for a few such sequences). Suppose that several feature points have been extracted from the moving objects in the two camera views of the scene. The task is to separate the trajectories of the feature points according to different motions. This application, which lies in the field of structure from motion
, is one of the fundamental problems in computer vision.
Given a physical point and its image correspondences in the two views , one can always form a joint image sample . It is shown in  that, under perspective camera projection, all the joint image samples corresponding to different motions live on different manifolds in , some having dimension 2 and others having dimension 4. Exploratory analysis applied to these data suggests that the manifolds in this dataset mostly have dimension 2 (see Figure 17). Therefore, we will apply our algorithm (HOSC) with to these data sets in order to compare with pairwise spectral clustering (SC-NJW, SC-LS).
We use the following parameter values for the two algorithms. In HOSC, we choose , while in SC we try both searching the interval (SC-NJW) and local scaling with at most 24 nearest neighbors (SC-LS).
The original data contains some outliers. In fact, 10 sequences out of the 13 are corrupted with outliers, with the largest percentage being about 32%. We first manually remove the outliers from those sequences and solely focus on the clustering aspects of the two algorithms. Next, we add outliers back and compare them regarding outliers removal. (Note that we need to provide both algorithms with the true percentage of outliers in each sequence.) By doing so we hope to evaluate the clustering and outliers removal aspects of an algorithm separately and thus in the most accurate way.
|Data||Clustering Errors||# True Outliers Detected|
Table 1 presents the results from the experiments above. Observe that HOSC achieved excellent clustering results in all but two sequences, with zero error on eight sequences, one mistake on sequence (13), and two mistakes on each of (1) and (4). We remark that HOSC also outperformed the algorithms in [10, Table 1], in terms of clustering accuracy, but due to the main aim of this paper, we do not include those results in Table 1. In contrast, each of SC-NJW and SC-LS failed on at least five sequences (with over 15% misclassification rates), both containing the two bad sequences for HOSC. As a specific example, we display in Figure 18 the clusters obtained by both HOSC and SC on sequence (7), demonstrating again that higher order affinities can significantly improve over pairwise affinities in the case of manifold data. Regarding outliers removal, HOSC is also consistently better than SC-NJW and SC-LS (if not equally good).
4.1 When the Underlying Surfaces Self-Intersect
In our generative model described in Section 1.2 we assume that the surfaces are submanifolds, implying that they do not self-intersect. This is really for convenience as there is essentially no additional difficulty arising from self-intersections. If we allow the surfaces to self-intersect, then we bound the maximum curvature (from above) and not the reach. We could, for example, consider surfaces of the form , where is locally bi-Lipschitz and has bounded second derivative. A similar model is considered in  in the context of set estimation. Clearly, proving that each cluster is connected in the neighborhood graph in this case is the same. The only issue is in situations where a surface comes within distance from another surface at a location where the latter intersects itself. The geometry involved in such a situation is indeed complex. If we postulate that no such situation arises, then our results generalize immediately to this setting.
4.2 When the Underlying Surfaces Have Boundaries
When the surfaces have boundaries, points near the boundary of a surface may be substantially connected with points on a nearby surface. See Figure 19 for an illustration. This is symptomatic of the fact that the algorithm is not able to resolve intersections in general, as discussed in Section 2.3, with the notable exception of clusters of dimension , as illustrated in the ‘two moons’ example of Figure 13.
If we require a stronger separation between the boundary of a surface and the other surfaces, specifically,
with , no point near the boundary of a cluster is close to a point from a different cluster. (A corresponding requirement in the context of outliers would be that outliers be separated from the boundary of a cluster by at least , with .)
4.3 When the Data is of Mixed Dimensions
In a number of situations, the surfaces may be of different intrinsic dimensions. An important instance of that is the study of the distribution of galaxies in space, where the galaxies are seen to cluster along filaments () and walls () . We propose a top-down approach, implementing HOSC for each dimension starting at and ending at (or between any known upper and lower bounds for ).
At each step, the algorithm is run on each cluster obtained from the previous step, including the set of points identified as outliers. Indeed, when the dimension parameter of the algorithm is set larger than the dimension of the underlying surfaces, HOSC may not be able to properly separate clusters. For example, two parallel segments satisfying the separation requirement of Theorem 1 still belong to a same plane and HOSC with dimension parameter would not be able to separate the two line segments. Another reason for processing the outlier bin is the greater disparity in the degrees of the data points in the neighborhood graph often observed with clusters of different dimensions. At each step, the number of clusters is determined automatically according to the procedure described in Section 2.1, for such information is usually not available. The parameters and are chosen according to (15). Partial results suggest that, under some additional sampling conditions, this top-down procedure is accurate under weaker separation requirements than required by pairwise methods, which handle the case of mixed dimensions seamlessly . The key is that an actual cluster , as defined in Section 1.2, is never cut into pieces. Indeed, properties (A1) and (A4) in the proof of Theorem 1, which guarantee the connectivity and regularity (in terms of comparable degrees) of the subgraph represented by , are easily seen to also be valid when the dimension parameter of the algorithm is set larger than . (This observation might explain the success of the SCC algorithm of  in some mixed settings when using an upper bound on the intrinsic dimensions.)
4.4 Clustering Based on Local Polynomial Approximations
For and an integer , let be the subclass of of -dimensional submanifolds such that, for every with tangent , the orthogonal projection is a -diffeomorphism with all partial derivatives of order up to bounded in supnorm by . For example, includes a subclass of surfaces of the form , where is locally bi-Lipschitz and has its first derivatives bounded. (We could also consider surfaces of intermediate, i.e., Hölder smoothness, a popular model in function and set estimation [18, 38].)
Given that surfaces in are well-approximated locally by polynomial surfaces, it is natural to choose an affinity based on the residual of the best -dimensional polynomial approximation of degree at most to a set of points . This may be implemented via the “kernel trick” with a polynomial kernel, as done in  for the special case of algebraic surfaces. The main difference with the case of surfaces that we consider in the rest of the paper is the degree of approximation to a surface by its osculating algebraic surface of order ; within a ball of radius , it is of order .
Partial results suggest that, under similar conditions, the kernel version of HOSC with known may be able to operate under a separation of the form (9), with the exponent replaced by and, in the presence of outliers, within a logarithmic factor of the best known sampling rate ratio achieved by any detection method [5, 4]:
Regarding the estimation of , defining the correlation dimension using the underlying affinity defined here allows to estimate accurately down to (essentially) , if the surfaces are all in . The arguments are parallel and we omit the details.
Thus, using the underlying affinity defined here may allow for higher accuracy, if the surfaces are smooth enough. However, this comes with a larger computational burden and at the expense of introducing a new parameter , which would need to be estimated if unknown, and we do not know a good way to do that.
4.5 Other Extensions
The setting we considered in this paper, introduced in Section 1.2, was deliberately more constrained than needed for clarity of exposition. We list a few generalizations below, all straightforward extensions of our work.
Sampling. Instead of the uniform distribution, we could use any other distribution with a density bounded away from and
, or with fast decaying tails such as the normal distribution.
Kernel. The rate of decay of the kernel dictates the range of the affinity (3). Let be a non-decreasing sequence such that For a compactly supported kernel, , while for the heat kernel, we can take . As we will take , is essentially supported in so that points that are further than apart have basically zero affinity. Specifically, we use the following bounds:
The results are identical, except that statements of the form are replaced with .
Measure of flatness. As pointed out in the introduction, any reasonable measure of linear approximation could be used instead. Our choice was driven by convenience and simplicity.
Appendix A Preliminaries
We gather here some preliminary results. Recall that, for , ; ; . For