On uniform consistency of spectral embeddings

In this section, we state our results in an informal manner. These results are made precise and proved in subsequent sections.

Our first main result concerns the effect of perturbation on the invariant subspace of an operator. It serves as a general recipe for establishing uniform consistency type results. Although in statistics and machine learning, we mainly work with real-valued functions, our main spectral perturbation result is stated for complex-valued functions. This choice is technically convenient because the complex numbers are algebraically closed while the real numbers are not. In most applications of the result, the complex-valued functions only take real values.

Suppose $\mathcal{H}$ is a complex Hilbert space whose elements are bounded complex-valued continuous functions over a domain $\mathcal{X}$. Let $T,\widetilde{T}$ be two operators from $\mathcal{H}$ to $\mathcal{H}$ that are close in Hilbert-Schmidt norm. Let $\{f_{i}\}_{i=1}^{K}$ be the top $K$ eigenfunctions of $T$ and $\{\tilde{f}_{i}\}_{i=1}^{K}$ be those of $\widetilde{T}$. As long as $\{f_{i}\}_{i=1}^{K}$ and $\{\tilde{f}_{i}\}_{i=1}^{K}$ are appropriately normalized, we expect $\{f_{i}\}_{i=1}^{K}$ to be close to $\{\tilde{f}_{i}\}_{i=1}^{K}$ up to some unitary transformation. This is indeed the case and is characterized as follows by our first result.

It is not hard to notice the correspondence between (1.4) and (1.3): $V_{1}$ is the analogue of $\Psi$; $\widetilde{V}_{1}$ is the analogue of $\Psi_{n}$; the two to infinity norm guarantees the convergence is uniform, and the distance metric $d$ is chosen to measure Euclidean distance up to a unitary transformation.¹¹1Normalized eigenfunctions of the same eigenvalue are only determined up to unitary transformation. This observation justifies naming $\inf\{\|V_{1}-\widetilde{V}_{1}Q\|_{2\to\infty}:Q\in\mathbb{U}^{K}\}$ the uniform consistency error. It is also worth mentioning that the constant $C_{2}$ is inversely proportional to a measure of the eigengap between the $K$-th and $K+1$-th eigenvalues of $T$. This provides further justification for studying the convergence of the leading eigenspace as a whole, rather than studying the convergence of the individual eigenspaces like in von Luxburg, Belkin and Bousquet [41]. Not only is the former more general and realistic, but it leads to better constants as well. In many applications, the top eigenvalues are usually clustered together, but there is a large gap between the top eigenvalues and the rest of the spectrum. Thus it is hard to estimate the corresponding eigenfunctions individually, but it is easy to estimate them altogether up to a unitary transformation.

Result 1 provides a general approach to proving uniform consistency: we simply need to bound the difference between the sample level operator and its population counterpart in an appropriate norm. The proof of Result 1 is also interesting in its own right. We identify the invariant subspace directly by solving an operator equation and appeal to the Newton-Kantorovich Theorem to characterize the solution. The main benefit of this approach is it overcomes the limitations of traditional approaches when working with non-unitarily invariant norms.

Our second main result is a finite sample uniform error bound for embedding in normalized spectral clustering. Let $C_{b}(\mathcal{X})$ denote the space of bounded continuous complex-valued functions over $\mathcal{X}$. Define $V_{1}:\mathbb{C}^{K}\to C_{b}(\mathcal{X})$ as $V_{1}\alpha=\sum_{i=1}^{K}\alpha_{i}f_{i}$ where $f_{1},\dots,f_{K}$ are the leading real-valued eigenfunctions of the normalized Laplacian operator. Define $\widehat{V}_{n,1}:\mathbb{C}^{K}\to C_{b}(\mathcal{X})$ as $\widehat{V}_{n,1}\alpha=\sum_{i=1}^{K}\alpha_{i}\widehat{f}_{n,i}$ where $\widehat{f}_{n,i}$ are defined as in (1.2) and real-valued. Applying Result 1, we obtain

Despite the fact that that Result 2 is an application of Result 1, its proof is by no means simple. The main technical challenge is establishing concentration bounds for Hilbert-Schmidt operators. Result 2 suggests that the convergence rate, under appropriate conditions, is $\mathcal{O}(\frac{1}{\sqrt{n}})$ (modulo a log factor). Moreover, in the context of clustering, the notion of uniform consistency leads to stronger assurances about the correctness of the clustering output. For example, in spectral clustering, the points are clustered based on their embeddings. Uniform convergence implies the embeddings of all points are close to their population counterpart. As long as the error in the embeddings are small enough, it is possible to show that all points are correctly clustered. This is not possible if the embeddings only “converge in mean”: $\|V_{1}-\widehat{V}_{n,1}Q\|_{2\to L^{2}(\mathcal{X},\mathbb{P})}\to 0$.

Most closely related to our results are the work of von Luxburg, Belkin and Bousquet [41] and Rosasco, Belkin and Vito [29]. For normalized spectral clustering, von Luxburg, Belkin and Bousquet [41] proved the convergence of the eigenvalues and spectral projectors of the sample level operator to their population counterparts. They also established uniform convergence of eigenfunctions whose corresponding eigenvalue has multiplicity one to their population counterparts. Our results are in the same vein as theirs in that we also study uniform convergence of eigenfunctions. We improve upon their uniform convergence result by considering multiple eigenfunctions at once and allowing for non-simple eigenvalues. In the context of unnormalized spectral clustering, Rosasco, Belkin and Vito [29] studied the convergence rate of the $l^{2}$-distance between the ordered spectrum of the sample level operator and that of the population operator and derived finite sample bound for the deviation between the sample level and population level spectral projections associated with the top $K$ eigenvalues. They also obtained finite sample spectral projection error bound for asymmetric normalized graph Laplacian. Our work is related to theirs because both study the convergence of the leading eigenspace and we owe much of our concentration results to them. The two works are also very distinct at the same time. Firstly, our notion of convergence is in uniform consistency of the eigenfunction, theirs is in terms of the induced RKHS norm between the spectral projectors. To the best of our knowledge, it is non-trivial to establish one set of results from the other. Secondly, we study the normalized symmetric graph Laplacian, while they study the unnormalized graph Laplacian and asymmetric normalized graph Laplacian.

The general relationship between the spectral properties of an empirical operator/matrix and that of its population counterpart has also been studied under other contexts. In Koltchinskii and Giné [22], it is proved that the ordered spectra of an integral operator and its empirical version tend to zero in $l^{2}$-distance almost surely if and only if the kernel is square integrable. Convergence rate and distributional limits were also obtained under stronger conditions. In Koltchinskii [21], the authors extended their own result by proving law of large numbers and central limit theorems for quadratic forms induced by spectral projections. The investigation of spectrum convergence is continued in Mendelson and Pajor [25] and Mendelson and Pajor [26], where the authors associated various types of distance metric between two ordered spectra to the deviation of the sample mean of i.i.d rank one operators from its population mean. Similar problems have also been studied in kernel principal component analysis (KPCA) literature. For example, in Shawe-Taylor et al. [31] and Blanchard, Bousquet and Zwald [5], the concentration property of the sum of the top $K$ eigenvalues and the sum of all but the top $K$ eigenvalues of the empirical kernel matrix are studied, because such partial sums are closely related to the reconstruction error of KPCA. In Zwald and Blanchard [42], a finite sample error bound on the difference between the projection operator to the leading eigenspace of the empirical covariance operator and that to the leading eigenspace of the population covariance operator is derived. We remark that none of the results mentioned in this paragraph addressed the consistency of the embedding directly, nor did any consider kernel matrix normalized by the degree matrix.

Unlike our results which are model-agnostic, the property of spectral methods has also been studied in model-specific settings. For example, Rohe, Chatterjee and Yu [28] and Lei and Rinaldo [23] investigated the spectral convergence properties of the graph Laplacian and the consistency of spectral clustering in terms of community membership recovery under stochastic block models. When the data are sampled from a finite mixture of nonparametric distributions, Shi, Belkin and Yu [32] studied how the leading eigenfunctions and eigenvectors of the population level integral operator can reflect clustering information; Schiebinger, Wainwright and Yu [30] studied the geometry of the embedded samples generated by normalized spectral clustering and showed that the embedded samples for different clusters are approximately orthogonal when the mixtures have small overlapping and the sample size is large. We remark that in all the results mentioned in this section so far, the kernel function is fixed. For the relationship between the graph Laplacian and the Laplace-Beltrami operator on a manifold and the properties of spectral clustering when the kernel is chosen adaptively, we refer readers to the series of work by Trillos et. al. [38, 39, 37] and the references therein.

Lastly, entrywise or row-wise analysis for eigevectors and eigenspaces of matrices has been studied in recent literature. For general purpose, deterministic $\ell_{2\to\infty}$ bounds are derived by Fan, Wang and Zhong [16], Cape, Tang and Priebe [7], Damle and Sun [12], where the first two are for rectangular matrices and the last one is for symmetric matrices. When probabilistic assumptions are imposed on the true and perturbed matrices, Cape, Tang and Priebe [8], Abbe et al. [1], Mao, Sarkar and Chakrabarti [24] obtain stronger $\ell_{2\to\infty}$ bounds for various tasks by taking advantages of the structure of the random matrices. Comparing to these literature, we remark that our work provides a deterministic bound which aids in the $\ell_{2\to\infty}$ perturbation theory of linear operators, and the bound can be applied to many problems in statistics (e.g., spectral clustering and kernel PCA) for helping characterize the spectral embedding of individual samples.

We view our main contributions as three fold and list them in the order of appearance. First, we demonstrate that the Newton-Kantorovich Theorem provides a general approach to studying the effect of local perturbations on the invariant spaces of an operator. This result may be of independent interest to researchers working on spectral perturbation theory. Second, we study the convergence of the embeddings via uniform consistency error and offer a general recipe for establishing non-asymptotic uniform consistency type results that handles multiple eigenfunctions at once and is not limited to simple eigenvalues. Third, we apply our recipe to normalized spectral clustering and give a novel proof of finite sample error bound on the uniform consistency error of the spectral embeddings.

The rest of the paper is organized as follows: A review of relevant mathematical preliminaries is provided in Section 2; the exact statement and proof for Result 1 is in section 3; the exact statement and proof for Result 2 is in section 4; a discussion of various issues relevant to our results is in section 5; proofs of some secondary lemmas and an additional application are relegated to the appendix.

We assume readers are familiar with basic concepts such as Banach spaces, Hilbert spaces, linear operators, operator norms, and spectra of operators. From now on, we let $\mathbb{K}$ denote either the field of real numbers or the field of complex numbers, $\mathcal{Y}_{1},\mathcal{Y}_{2}$ denote Banach spaces over the same field $\mathbb{K}$, and $\mathcal{H}_{1},\mathcal{H}_{2}$ denote Hilbert spaces over the same field $\mathbb{K}$.

We would like to first highlight a nuance in the definition of linear operator. For a linear operator $A:\mathcal{Y}_{1}\to\mathcal{Y}_{2}$, we adopt the convention from Kato [20] and allow $A$ to be defined only on a linear manifold in $\mathcal{Y}_{1}$,²²2In Kato [20], linear manifold is just a synonym for affine subspace. denoted $D(A)$.³³3In Ciarlet [9], for example, such a distinction isn’t made. We call $D(A)$ the domain of $A$ and can naturally define the range of $A$ as $R(A):=\{Ay\;|\;y\in D(A)\}$. As for $\mathcal{Y}_{1},\mathcal{Y}_{2}$, we call them the domain space and the range space respectively.

For a linear operator $A:\mathcal{Y}_{1}\to\mathcal{Y}_{2}$, we say $A$ bounded if $\sup_{\|y\|_{\mathcal{Y}_{1}}=1}\|Ay\|_{\mathcal{Y}_{2}}<\infty$, and when $A$ is bounded, we define its operator norm $\|A\|:=\sup_{\|y\|_{\mathcal{Y}_{1}}=1}\|Ay\|_{\mathcal{Y}_{2}}$. Throughout the paper, when $\|\cdot\|$ has no subscript, it defaults to operator norm. We use $\mathcal{L}(\mathcal{Y}_{1},\mathcal{Y}_{2})$ to denote the space of all bounded linear operators from $\mathcal{Y}_{1}$ to $\mathcal{Y}_{2}$. When $\mathcal{Y}_{1}=\mathcal{Y}_{2}$, we simply write $\mathcal{L}(\mathcal{Y}_{1},\mathcal{Y}_{1})$ as $\mathcal{L}(\mathcal{Y}_{1})$. We say $A$ is a compact operator if the closure of the image of any bounded set in $\mathcal{Y}_{1}$ under $A$ is compact. It is know that compact operators are bounded.

For a bounded linear operator $A:\mathcal{H}_{1}\to\mathcal{H}_{2}$, define its adjoint $A^{*}:\mathcal{H}_{2}\to\mathcal{H}_{1}$ as the unique operator from $\mathcal{H}_{2}$ to $\mathcal{H}_{1}$ satisfying $\langle Af,g\rangle_{\mathcal{H}_{2}}=\langle f,A^{*}g\rangle_{\mathcal{H}_{1}}$ for $\forall f\in\mathcal{H}_{1},\forall g\in\mathcal{H}_{2}$. Here, we use $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ to denote the inner product in the Hilbert space $\mathcal{H}$. A basic property of $A^{*}$ is $\|A\|_{\mathcal{H}_{1}\to\mathcal{H}_{2}}=\|A^{*}\|_{\mathcal{H}_{2}\to\mathcal{H}_{1}}$, where the norm is operator norm and we use the $\mathcal{H}_{1}\to\mathcal{H}_{2}$ notation to explicitly specify the domain space and range space. When $\mathcal{H}_{1}=\mathcal{H}_{2}$, $A$ is called self-adjoint if $A$ is equal to its adjoint $A^{*}$, and $A$ is called positive if for any $f\in\mathcal{H}_{1}$, $\langle Af,f\rangle_{\mathcal{H}_{1}}\geq 0$.

We say a Hilbert space is separable if it has a basis of countably many elements. We say a bounded linear operator $A:\mathcal{H}_{1}\to\mathcal{H}_{2}$ is Hilbert-Schmidt if $\sum_{i\in\mathcal{I}}\|Ae_{i}\|_{\mathcal{H}_{2}}^{2}<\infty$ where $\{e_{i}:i\in\mathcal{I}\}$ is an orthonormal basis of $\mathcal{H}_{1}$. We use $HS(\mathcal{H}_{1},\mathcal{H}_{2})$ to denote the space of all Hilbert-Schmidt operators from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$; this space is also a Hilbert space with respect to the inner product $\langle A,B\rangle_{HS}:=\sum_{i\in\mathcal{I}}\langle Ae_{i},Be_{i}\rangle_{\mathcal{H}_{2}}$. We use $\|\cdot\|_{HS}$ to denote the norm induced by this inner product and note that all Hilbert-Schmidt operators are compact. We also note the Hilbert-Schmidt norm is stronger than operator norm in that $\|A\|\leq\|A\|_{HS}$, and the Hilbert-Schmidt norm is compatible with the operator norm in the following sense: for any Hilbert-Schmidt operator $A$ and bounded operator $B$, their product $AB$ and $BA$ are Hilbert-Schmidt and their Hilbert-Schmidt norm satisfies

In this subsection, we set $\mathbb{K}=\mathbb{C}$. Let $A:\mathcal{H}_{1}\to\mathcal{H}_{1}$ be a bounded linear operator. Similar to matrices, we say $\lambda\in\mathbb{C}$ is an eigenvalue of $A$ if for some eigenvector $f\in\mathcal{H}_{1}$,

In other words, $\lambda$ is an eigenvalue if the null space $N(\lambda I-A)$ is not $\{0\}$. We call $N(\lambda I-A)$ the eigenspace associated with $\lambda$, and the dimension of $N(\lambda I-A)$ is called the geometric multiplicity of $\lambda$. The spectrum of $A$ is defined as $\sigma(A):=\mathbb{C}\setminus\rho(A)$, where $\rho(A)$ is the resolvent set

Eigenvalues are in the spectrum, but $\sigma(A)$ generally contains more than just eigenvalues. If $A$ is a compact operator, $\sigma(A)$ has the following structure: $\sigma(A)\setminus\{0\}$ is a countable set of isolated eigenvalues, each with finite geometric multiplicity, and the only possible accumulation point of $\sigma(A)$ is $0$. If $A$ is self-adjoint, then all the eigenvalues must be real. If $A$ is a positive operator, then all its eigenvalues are real and non-negative. Therefore for any compact positive self-adjoint operator, we can arrange the non-zero eigenvalues of $A$ into a non-increasing sequence of positive numbers,⁴⁴4Because the largest eigenvalue is bounded by the operator norm of $A$. and repeat each eigenvalue for a number of times equal to its geometric multiplicity.

Another remarkable fact in the spectral theory of linear operators concerns spectral projection. Let $\Gamma\subset\rho(A)$ be a closed simple rectifiable curve. Assume the part of $\sigma(A)$ enclosed inside $\Gamma$ is a finite number of eigenvalues $\lambda_{1},\lambda_{2},\ldots,\lambda_{K}$. Then the projection $P:\mathcal{H}_{1}\to\mathcal{H}_{1}$ which projects to the direct sum of the eigenspaces of $\{\lambda_{i}\}_{i=1}^{K}$, i.e. $\bigoplus_{i=1}^{K}N(\lambda_{i}I-A)$, can be defined. Technicalities aside, this projection has the following contour integration expression

Let $\mathcal{X}$ be a bounded open subset of $\mathbb{R}^{p}$, we now define several function spaces we are going to work with. Define the space of bounded continuous functions $C_{b}(\mathcal{X})$ as

It can be shown that $\|f\|_{\infty}:=\sup_{x\in\mathcal{X}}|f(x)|$ is a norm on $C_{b}(\mathcal{X})$ and $C_{b}(\mathcal{X})$ is a Banach space with respect to this infinity norm.

We can also define the space of complex-valued square integrable functions $L^{2}(\mathcal{X},\mu)$. Suppose $(\mathcal{X},\mathcal{B},\mu)$ is a measure space where $\mathcal{B}$ is the Lebesgue $\sigma$-algebra and $\mu$ is a measure, then $L^{2}(\mathcal{X},\mu)$ is defined as the set of measurable functions such that

In fact, $L^{2}(\mathcal{X},\mu)$ is a Hilbert space with respect to the inner product

We also define $l^{2}$, the space of square summable infinite sequence of complex numbers. It is well known that $l^{2}$ is a complex Hilbert space with respect to the inner product

Let $\mathcal{X}$ be a subset of $\mathbb{R}^{p}$ and $\mathcal{H}$ be a set of functions $f:\mathcal{X}\to\mathbb{C}$. Suppose $\mathcal{H}$ is a Hilbert space with respect to some inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}}$. If in $\mathcal{H}$, all point evaluation functionals are bounded, i.e.

where $C_{x}$ is some constant depending on $x$, then it can be shown that there exists a unique conjugate symmetric positive definite kernel function $k:\mathcal{X}\times\mathcal{X}\to\mathbb{C}$, such that the following reproducing property is satisfied:

The kernel $k$ is called the reproducing kernel and $\mathcal{H}$ is called a reproducing kernel Hilbert space (RKHS).

We say a kernel function $k$ is positive definite if for any $n\in\mathbb{N}^{+}$, any $x_{1},x_{2},\ldots,x_{n}\in\mathcal{X}$ and any $\xi_{1},\xi_{2},\ldots,\xi_{n}\in\mathbb{C}$, the quadratic form

is non-negative. The kernel function for any RKHS is positive definite.

In this section, our goal is to find a $Y\in\mathcal{L}(\mathbb{C}^{K},\tilde{l}^{2})$ such that the range of $V_{1}+V_{2}Y\in\mathcal{L}(\mathbb{C}^{K},\mathcal{H})$ is an invariant subspace of $\widetilde{T}$. It turns out that any $Y$ that satisfies the following quadratic operator equation suffices.

After characterizing the invariant subspace of $\widetilde{T}$ in terms of a solution of (3.5), we apply the Newton-Kantorovich Theorem to prove a solution to (3.5) exists. The Newton-Kantorovich Theorem constructs a root of a function between Banach spaces when certain conditions on the function itself and its first and second order derivatives are met. The construction is algorithmic: the root is the limit point of a sequence of iterates generated by the Newton-Raphson method for root finding. The exact version of the Newton-Kantorovich Theorem we use is from the appendix of Karow and Kressner [19].

We are now ready to prove the proposition below, which states that when $\|E_{11}\|_{HS}$, $\|E_{21}\|_{HS}$, $\|E_{22}\|_{HS}$, $\|E_{12}\|_{HS}$ are small relative to $\text{sep}(T_{11},T_{22})$, equation (3.5) has a solution.