Topological Signal Processing over
Simplicial Complexes

Given a finite set $\mathcal{V}\triangleq\{v_{0},\ldots,v_{N-1}\}$ of $N$ points (vertices), a $k$-simplex $\sigma^{k}_{i}$ is an unordered set $\{v_{i_{0}},\ldots,v_{i_{k}}\}$ of $k+1$ points with $0\leq i_{j}\leq N-1$, for $j=0,\ldots,k$, and $v_{i_{j}}\neq v_{i_{n}}$ for all $i_{j}\neq i_{n}$. A face of the $k$-simplex $\{v_{i_{0}},\ldots,v_{i_{k}}\}$ is a $(k-1)$-simplex of the form $\{v_{i_{0}},\ldots,v_{i_{j-1}},v_{i_{j+1}},\ldots,v_{i_{k}}\}$, for some $0\leq j\leq k$. Every $k$-simplex has exactly $k+1$ faces. An abstract simplicial complex ${\cal X}$ is a finite collection of simplices that is closed under inclusion of faces, i.e., if ${\cal\sigma}_{i}\in{\cal X}$, then all faces of $\sigma_{i}$ also belong to ${\cal X}$. The order (or dimension) of a simplex is one less than its cardinality. Then, a vertex is a 0-dimensional simplex, an edge has dimension $1$, and so on. The dimension of a simplicial complex is the largest dimension of any of its simplices. A graph is a particular case of an abstract simplicial complex of order $1$, containing only simplices of order $0$ (vertices) and $1$ (edges).

If the set of points is embedded in a real space $\mathbb{R}^{D}$ of dimension $D$, we can associate a geometric simplicial complex with the abstract complex. A set of points in a real space $\mathbb{R}^{D}$ is affinely independent if it is not contained in a hyperplane; an affinely independent set in $\mathbb{R}^{D}$ contains at most $D+1$ points. A geometric $k$-simplex is the convex hull of a set of $k+1$ affinely independent points, called its vertices. Hence, a point is a $0$-simplex, a line segment is a $1$-simplex, a triangle is a $2$-simplex, a tetrahedron is a $3$-simplex, and so on. A geometric simplicial complex shares the fundamental property of an abstract simplicial complex: It is a collection of simplices that is closed under inclusion and with the further property that the intersection of any two simplices in ${\cal X}$ is also a simplex in ${\cal X}$, assuming that the empty set is an element of every simplicial complex. Although geometric simplicial complexes are easier to visualize and, for this reason, we will often use geometric terms like edges, triangles, and so on, as synonyms of pairs, triplets, we do not require the simplicial complex to be embedded in any real space, so as to leave the treatment as general as possible.

The structure of a simplicial complex is captured by the neighborhood relations of its subsets. As with graphs, it is useful to introduce first the orientation of the simplices. Every simplex, of any order, can have only two orientations, depending on the permutations of its elements. Two orientations are equivalent, or coherent, if each of them can be recovered from the other by an even number of transpositions, where each transposition is a permutation of two elements [10]. A $k$-simplex $\sigma_{i}^{k}\equiv\{v_{i_{0}},v_{i_{1}},\ldots,v_{i_{k}}\}$ of order $k$, together with an orientation is an oriented $k$-simplex and is denoted by $\left[v_{i_{0}},v_{i_{1}},\ldots,v_{i_{k}}\right]$. Two simplices of order $k$, $\sigma_{i}^{k},\sigma_{j}^{k}\in{\cal X}$, are upper adjacent in ${\cal X}$, if both are faces of a simplex of order $k+1$. Two simplices of order $k$, $\sigma_{i}^{k},\sigma_{j}^{k}\in{\cal X}$, are lower adjacent in ${\cal X}$, if both have a common face of order $k-1$ in ${\cal X}$. A $(k-1)$-face $\sigma_{j}^{k-1}$ of a $k$-simplex $\sigma_{i}^{k}$ is called a boundary element of $\sigma_{i}^{k}$. We use the notation $\sigma_{j}^{k-1}\subset\sigma_{i}^{k}$ to indicate that $\sigma_{j}^{k-1}$ is a boundary element of $\sigma_{i}^{k}$. Given a simplex $\sigma_{j}^{k-1}\subset\sigma_{i}^{k}$, we use the notation $\sigma_{j}^{k-1}\sim\sigma_{i}^{k}$ to indicate that the orientation of $\sigma_{j}^{k-1}$ is coherent with that of $\sigma_{i}^{k}$, whereas we write $\sigma_{j}^{k-1}\nsim\sigma_{i}^{k}$ to indicate that the two orientations are opposite.

For each $k$, $C_{k}({\cal X},\mathbb{R})$ denotes the vector space obtained by the linear combination, using real coefficients, of the set of oriented $k$-simplices of ${\cal X}$. In algebraic topology, the elements of $C_{k}({\cal X},\mathbb{R})$ are called $k$-chains. If $\{\sigma_{1}^{k},\ldots,\sigma_{n_{k}}^{k}\}$ is the set of $k$-simplices in ${\cal X}$, a $k$-chain $\tau_{k}$ can be written as $\tau_{k}=\sum_{i=1}^{n_{k}}\alpha_{i}\sigma_{i}^{k}$. Then, given the basis $\{\sigma_{1}^{k},\ldots,\sigma_{n_{k}}^{k}\}$, a chain $\tau_{k}$ can be represented by the vector of its expansion coefficients $(\alpha_{1},\ldots,\alpha_{n_{k}})$. An important operator acting on ordered chains is the boundary operator. The boundary of the ordered $k$-chain $[v_{i_{0}},\ldots,v_{i_{k}}]$ is a linear mapping $\partial_{k}:C_{k}({\cal X},\mathbb{R})\rightarrow C_{k-1}({\cal X},\mathbb{R})$ defined as

So, for example, given an oriented triangle $\sigma^{2}_{i}\triangleq[v_{i_{0}},v_{i_{1}},v_{i_{2}}]$, its boundary is

i.e., a suitable linear combination of its edges. It is straightforward to verify, by simple substitution, that the boundary of a boundary is zero, i.e., $\partial_{k}\partial_{k+1}=0$.

It is important to remark that an oriented simplex is different from a directed one. As with graphs, an oriented edge establishes which direction of the flow is considered positive or negative, whereas a directed edge only permits flow in one direction [42]. In this work we will considered oriented, undirected simplices.

The structure of a simplicial complex ${\cal X}$ of dimension $K$, shortly named $K$-simplicial complex, is fully described by the set of its incidence matrices $\mathbf{B}_{k}$, $k=1,\ldots,K$. Given an orientation of the simplicial complex ${\cal X}$, the entries of the incidence matrix $\mathbf{B}_{k}$ establish which $k$-simplices are incident to which $(k-1)$-simplices. Then $\mathbf{B}_{k}$ is the matrix representation of the boundary operator. Formally speaking, its entries are defined as follows:

If we consider, for simplicity, a simplicial complex of order two, composed of a set ${\cal V}$ of vertices, a set ${\cal E}$ of edges, and a set ${\cal T}$ of triangles, having cardinalities $V=|{\cal V}|$, $E=|{\cal E}|$, and $T=|{\cal T}|$, respectively, we need to build two incidence matrices $\mbox{$\mathbf{B}$}_{1}\in\mathbb{R}^{V\times E}$ and $\mbox{$\mathbf{B}$}_{2}\in\mathbb{R}^{E\times T}$.

From (1), the property that the boundary of a boundary is zero maps into the following matrix form

The structure of a $K$-simplicial complex is fully described by its high order combinatorial Laplacian matrices [43], of order $k=0,\ldots,K$, defined as

It is worth emphasizing that all Laplacian matrices of intermediate order, i.e. $k=1,\ldots,K-1$, contain two terms: The first term, also known as lower Laplacian, expresses the lower adjacency of $k$-order simplices; the second terms, also known as upper Laplacian, expresses the upper adjacency of $k$-order simplices. So, for example, two edges are lower adjacent if they share a common vertex, whereas they are upper adjacent if they are faces of a common triangle. Note that the vertices of a graph can only be upper adjacent, if they are incident to the same edge. This is why the graph Laplacian $\mathbf{L}_{0}$ contains only one term.

There are interesting relations between the eigenvectors of Laplacian matrices of different order [45], [46] which is useful to recall as they play a key role in spectral analysis. The following properties hold true for the eigendecomposition of $k$-order Laplacian matrices, with $k=1,\ldots,K-1$.

Let us consider the eigendecomposition of the $k$-th order Laplacian, for $k=1,\ldots,K-1$,

The structure of $\mathbf{L}_{k}$, together with the property $\mathbf{B}_{k}\mathbf{B}_{k+1}=\mathbf{0}$, induces an interesting decomposition of the space $\mathbb{R}^{D_{k}}$ of signals of order $k$ of dimension ${D_{k}}$. First of all, the property $\mathbf{B}_{k}\mathbf{B}_{k+1}=\mathbf{0}$ implies ${\rm img}(\mathbf{B}_{k+1})\subseteq{\rm ker}(\mathbf{B}_{k})$. Hence, each vector $\mbox{\boldmath$x$}\in{\rm ker}(\mathbf{B}_{k})$ can be decomposed into two parts: one belonging to ${\rm img}(\mathbf{B}_{k+1})$ and one orthogonal to it. Furthermore, since the whole space $\mathbb{R}^{D_{k}}$ can always be written as $\mathbb{R}^{D_{k}}\equiv{\rm ker}(\mathbf{B}_{k})\oplus{\rm img}(\mathbf{B}_{k}^{T})$, it is possible to decompose $\mathbb{R}^{D_{k}}$ into the direct sum [47]

where the vectors in $\text{ker}(\mathbf{L}_{k})$ are also in $\text{ker}(\mathbf{B}_{k})$ and $\text{ker}(\mathbf{B}_{k+1}^{T})$. This implies that, given any signal $\mbox{\boldmath$s$}^{k}$ of order $k$, there always exist three signals $\mbox{\boldmath$s$}^{k-1}$, $\mbox{\boldmath$s$}_{H}^{k}$, and $\mbox{\boldmath$s$}^{k+1}$, of order $k-1$, $k$, and $k+1$, respectively, such that $\mbox{\boldmath$s$}^{k}$ can always be expressed as the sum of three orthogonal components:

This decomposition is known as Hodge decomposition [48] and it is the extension of the Hodge theory for differential forms on Riemannian manifold to simplicial complexes. The subspace $\text{ker}(\mathbf{L}_{k})$ is called harmonic subspace since each $\mbox{\boldmath$s$}_{H}^{k}\in\text{ker}(\mathbf{L}_{k})$ is a solution of the discrete Laplace equation

When embedded in a real space, a fundamental property of geometric simplicial complexes of order $k$ is that the dimensions of $\text{ker}(\mathbf{L}_{k})$, for $k=0,\ldots,K$ are topological invariants of the $K$-simplicial complex, i.e. topological features that are preserved under homeomorphic transformations of the space. The dimensions of $\text{ker}(\mathbf{L}_{k})$ are also known as Betti numbers $\beta_{k}$ of order $k$: $\beta_{0}$ is the number of connected components of the graph, $\beta_{1}$ is the number of holes, $\beta_{2}$ is the number of cavities, and so on [48].

The decomposition in (19) shows an interesting interplay between signals of different order, which we will exploit in the ensuing sections. Before proceeding, it is useful to clarify the meaning of the terms appearing in (19). Let us consider, for simplicity, the analysis of flow signals, i.e. the case $k=1$. To this end, it is useful to introduce the curl and divergence operators, in analogy with their continuous time counterpart operators applied to vector fields. More specifically, given an edge signal $\mbox{\boldmath$s$}^{1}$, the (discrete) curl operator is defined as

This operator maps the edge signal $\mbox{\boldmath$s$}^{1}$ onto a signal defined over the triangle sets, i.e. in $\mathbb{R}^{T}$, and it is straightforward to verify that the generic $i$-th entry of the $\mbox{curl}(\mbox{\boldmath$s$}^{1})$ is a measure of the flow circulating along the edges of the $i$-th triangle. As an example for the simplicial complex in Fig. 1, we have

and, defining $\mbox{\boldmath$s$}^{1}=[e_{1},e_{2},\ldots,e_{10},e_{11}]^{T}$, we get $\mbox{curl}(\mbox{\boldmath$s$}^{1})=[e_{3}-e_{4}+e_{5},e_{2}-e_{3}+e_{7},-e_{7}+e_{8}+e_{9}]^{T}$. Each entry of $\mbox{\boldmath$s$}^{1}$ is then the circulation over the corresponding triangle.

Similarly, the (discrete) divergence operator maps the edge signal $\mbox{\boldmath$s$}^{1}$ onto a signal defined over the vertex space, i.e. ${\mathbb{R}}^{V}$, and it is defined as

Again, by direct substitution, it turns out that the $i$-th entry of $\mbox{div}(\mbox{\boldmath$s$}^{1})$ represents the net-flow passing through the $i$-th vertex, i.e. the difference between the inflow and outflow at node $i$. Thus a non-zero divergence reveals the presence of a source or sink node. For the example in Fig. 1, we get

so that $\mbox{div}(\mbox{\boldmath$s$}^{1})=[-e_{1}-e_{6},e_{1}-e_{2}-e_{3}-e_{4},e_{2}-e_{7}-e_{9}-e_{10},e_{10}-e_{11},-e_{8}+e_{9}+e_{11},e_{3}-e_{5}+e_{7}+e_{8},e_{4}+e_{5}+e_{6}]^{T}$.
If we consider equation (19) in the case $k=1$,

recalling that $\mbox{$\mathbf{B}$}_{1}\mbox{$\mathbf{B}$}_{2}=\mathbf{0}$, it is easy to check that the first component in (24) has zero curl, and then it may be called an irrotational component, whereas the third component has zero divergence, and then it may be called a solenoidal component, in analogy to the calculus terminology used for vector fields. The harmonic component $\mathbf{s}_{H}^{1}$ is a flow vector that is both curl-free and divergence-free. Notice also that, in (24), $\mathbf{B}_{1}^{T}\,\mbox{\boldmath$s$}^{0}$ represents the (discrete) gradient of $\mbox{\boldmath$s$}^{0}$.

In this section, we propose a method to build a unitary basis to represent edge signals, reflecting topological properties of the complex, more specifically triple-wise relations. The idea underlying the method is to search a basis that minimizes a real-valued function capturing triple-wise relations. For example, in the graph case, a key topological property is connectivity, which is well captured by the cut function. The associated real-valued function can be built using the so called Lovász extension [49], [50] of the cut size.

More specifically, given a graph ${\cal G({\cal V},{\cal E})}$ and a partition of its vertex set ${\cal V}$ in two sets ${\cal A}_{0}$ and ${\cal A}_{1}$, with ${\cal A}_{0}\cup{\cal A}_{1}={\cal V}$ and ${\cal A}_{0}\cap{\cal A}_{1}=\emptyset$, the cut size is defined as

where $a_{ij}=1$ if $(i,j)\in{\cal{E}}$ and $a_{ij}=0$ otherwise. $F_{0}({\cal A}_{0},{\cal A}_{1})$ is a set function and is known to be a submodular function [49]. Now, we want to translate the set function $F_{0}({\cal A}_{0},{\cal A}_{1})$ onto a real-valued function $f_{0}(\mbox{\boldmath$x$}^{0})$, defined over ${\mathbb{R}}^{V}$, to be used for the subsequent optimization. This can be done using the so called Lovász extension [49], which is equal to:

with $\mbox{\boldmath$x$}^{0}\in\mathbb{R}^{V}$. The function in (26) measures the total variation of a signal defined over the nodes of a graph and then it can be used as a regularization function, whenever the signal of interest is known to be a smooth function. The function in (26) formed the basis of the method proposed in [51] to build a unitary basis for analyzing signals defined over the vertices of a graph. More specifically, in [51] the basis was built by solving the following optimization problem

In the above problem, the objective function to be minimized is convex, but the problem is non-convex because of the unitarity constraint. To simplify the search for the basis, the objective function in (27) can be relaxed to become

Substituting (28) in (27), we still have a non-convex problem. However, its solution is known to be given by the eigenvectors of $\mathbf{L}_{0}$. From this perspective, the basis typically used in the GFT, built as the eigenvectors of the Laplacian matrix, can be interpreted as the solution of the above optimization problem, after relaxation.
Now, we extend the previous approach to find a regularization function capturing triple-wise relations, useful to analyze signals defined over the edges of a graph. In the previous case, the analysis of node signals started from the bi-partition of a graph and then the construction of a real-valued extension of the cut size. Here, to analyze edge signals, we need to start from the tri-partition of the discrete set and define a set function counting the number of triangles gluing the three sets together. The function to be minimized should then be the real valued (Lovász) extension of such a set function.

The combinatorial study of simplicial complexes has attracted increasing attention and the generalization of Cheeger inequalities to simplicial complexes has been the subject of several works [52], [53]. In particular, Hein et al. introduced the total variation on hypergraphs as the Lovász extension of the hypergraph cut, defined as the size of the hyperedges set connecting a bipartition of the vertex set [54]. In this work, we derive a Lovász extension defined over the edge set. More formally, we study a simplicial complex ${\cal X}({\cal V},{\cal E},{\cal T})$ of order two, as in the example sketched in Fig.1, and consider the partition of ${\cal V}$ in three sets $({\cal A}_{0},{\cal A}_{1},{\cal A}_{2})$. The triple-wise coupling function is now defined as

where $a_{ijk}=1$ if $(i,j,k)\in{\cal{T}}$ and $a_{ijk}=0$ otherwise. Our main result, stated in the following theorem, is that the Lovász extension of the triple-wise coupling function gives rise to a measure of the curl of edge signals along triangles.

The function in (30) represents the sum of the absolute values of the curls over all the triangles. Differently from [54], where the total variation over a hypergraph was built from a bipartition of a discrete set and it was a function defined over $\mathbb{R}^{V}$, in our case, we start from a triparition of the original set and we build a real-valued extension, defined over $\mathbb{R}^{E}$, i.e. a space of dimension equal to the number of edges. A suitable unitary basis for representing (curling) edge signals can then be found as the solution of the following optimization problem

The objective function is convex, but the problem is non-convex because of the unitarity constraint. The above problem can be solved resorting to the algorithm proposed in [51], tuned to the new objective function given in (30). Alternatively, similarly to what is typically done with graphs, $f_{1}(\mbox{\boldmath$x$}^{1})$ can be relaxed and replaced with

Substituting this function back to (31), the solution is given by the eigenvectors of $\mbox{$\mathbf{B}$}_{2}\mbox{$\mathbf{B}$}_{2}^{T}$.

The above arguments show that the eigenvectors of $\mbox{$\mathbf{B}$}_{2}\mbox{$\mathbf{B}$}_{2}^{T}$ provide a suitable basis to analyze edge signals having a curling behavior. However, as we know from the Hodge decomposition recalled in the previous section, edge signals contain other useful components that are orthogonal to solenoidal signals, namely irrotational and harmonic components. Then, in general, it is useful to take as a unitary basis for analyzing edge signals all the eigenvectors of $\mbox{$\mathbf{L}$}_{1}$, i.e. the eigenvectors associated to the nonzero eigenvalues of $\mbox{$\mathbf{B}$}_{2}\mbox{$\mathbf{B}$}_{2}^{T}$ and of $\mbox{$\mathbf{B}$}_{1}^{T}\mbox{$\mathbf{B}$}_{1}$, plus the eigenvectors associated to the kernel of $\mbox{$\mathbf{L}$}_{1}$. In summary, the theory developed in this section provides a further argument, rooted on intrinsic topological properties of the simplicial complex on which the signal is defined, to exploit the Hodge decomposition to find a suitable basis for the analysis of edge signals. Furthermore, the theory says that using the Laplacian eigenvectors comes as a consequence of a relaxation step. In many cases, whenever the numerical complexity is not an issue, it may be better to keep the $\ell_{1}$-norm objective functions given in (26) and (30), as this would yield more sparse, and then more informative, representations, as a generalization of what done for directed graphs in [51].

In the following theorem [56] we provide a necessary and sufficient condition to recover the edge signal $\mbox{\boldmath$s$}^{1}$ from its samples $\mbox{\boldmath$s$}^{1}_{\mathcal{S}}\triangleq\mathbf{D}_{\mathcal{S}}\mbox{\boldmath$s$}^{1}$.

In words, the above conditions mean that there can be no $\mathcal{F}$-bandlimited signals that are perfectly localized on the complementary set $\bar{\mathcal{S}}$. Perfect recovery of the signal $\mbox{\boldmath$s$}^{1}$ from $\mbox{\boldmath$s$}^{1}_{\mathcal{S}}$ can be achieved as

where $\mathbf{Q}_{\mathcal{S}}=(\mathbf{I}-\bar{\mathbf{D}}_{\mathcal{S}}\mathbf{F}_{\mathcal{F}})^{-1}$. The existence of the above inverse is ensured by condition (47). In fact, the reconstruction error can be written as [56]

where we exploited in the second equality the bandlimited condition $\mbox{\boldmath$s$}^{1}=\mathbf{F}_{\mathcal{F}}\mbox{\boldmath$s$}^{1}$.

To make (47) holds true, we must guarantee that $\mathbf{D}_{\mathcal{S}}\mathbf{F}_{\mathcal{F}}\mbox{\boldmath$s$}^{1}\neq{\mathbf{0}}$ or, equivalently, that the matrix $\mathbf{D}_{\mathcal{S}}\mathbf{F}_{\mathcal{F}}$ is full column rank, i.e. $\text{rank}(\mathbf{D}_{\mathcal{S}}\mathbf{F}_{\mathcal{F}})=|\mathcal{F}|$. Then, a necessary condition to ensure this holds is $|\mathcal{S}|\geq|\mathcal{F}|$.
An alternative way to retrieve the overall signal $\mbox{\boldmath$s$}^{1}$ from its samples can be obtained as follows. If (47) holds true, the entire signal $\mbox{\boldmath$s$}^{1}$ can be recovered from $\mbox{\boldmath$s$}^{1}_{\mathcal{S}}$ as follows

where $\mathbf{U}_{\mathcal{F}}$ is the $E\times|\mathcal{F}|$ matrix whose columns are the eigenvectors of $\mbox{$\mathbf{L}$}_{1}$ associated with the frequency set $\mathcal{F}$.

In this section, we consider the case where we take samples of signals of different order and we propose two alternative strategies to retrieve an edge signal $\mbox{\boldmath$s$}^{1}$ from these samples.
The first approach aims at recovering $\mbox{\boldmath$s$}^{1}$ by using both, the vertex signal samples $\mbox{\boldmath$s$}^{0}_{\mathcal{A}}=\mathbf{D}_{\mathcal{A}}\mbox{\boldmath$s$}^{0}$, with $\mathcal{A}\subseteq\mathcal{V}$, and the edge samples $\mbox{\boldmath$s$}^{1}_{\mathcal{S}}=\mathbf{D}_{\mathcal{S}}\mbox{\boldmath$s$}^{1}$. Hereinafter, we denote by $\mathcal{F}_{\text{irr}}$, $\mathcal{F}_{\text{sol}}$ and $\mathcal{F}_{\text{H}}$ the set of frequency indexes in $\mathcal{F}$ corresponding to the eigenvectors of $\mbox{$\mathbf{L}$}_{1}$ belonging, respectively, to the irrotational, solenoidal and harmonic subspaces. Note that, if $\mbox{\boldmath$s$}^{1}$ is $\mathcal{F}$-bandlimited then also $\mbox{\boldmath$s$}^{1}_{H}$, $\mbox{\boldmath$s$}^{1}_{\text{sol}}$ and $\mbox{\boldmath$s$}^{1}_{\text{irr}}$ are bandlimited with bandwidth, respectively, $|\mathcal{F}_{\text{H}}|$, $|\mathcal{F}_{\text{sol}}|$ and $|\mathcal{F}_{\text{irr}}|$. Define $\mathcal{F}_{\text{sH}}$ as the set of frequency indexes such that $\mathcal{F}_{\text{sH}}=\mathcal{F}_{\text{H}}\cup\mathcal{F}_{\text{sol}}$. Furthermore, given the matrix $\mathbf{U}_{0}$ with columns the eigenvectors of $\mbox{$\mathbf{L}$}_{0}$, we define the operator $\mathbf{F}_{\mathcal{F}_{0}}^{0}=\mathbf{U}_{0}\mathbf{\Sigma}_{\mathcal{F}_{0}}\mathbf{U}^{T}_{0}$ where $|\mathcal{F}_{0}|$ denotes the bandwidth of $\mbox{\boldmath$s$}^{0}$. Let $\mathcal{C}_{1}$ be the set of indexes in $\mathcal{F}_{0}$ associated with the eigenvectors belonging to $\text{ker}(\mbox{$\mathbf{L}$}_{0})$ and denote by $\mbox{$\mathbf{U}$}^{0}_{\mathcal{F}_{0}-\mathcal{C}_{1}}$ the matrix whose columns are the eigenvectors of $\mbox{$\mathbf{L}$}_{0}$ associated with the frequency set $\mathcal{F}_{0}-\mathcal{C}_{1}$. Then, we can state the following theorem.