Out-of-Sample Extension of the Fuzzy Transform

This article addresses the definition and computation of the out-of-sample membership functions and the resulting out-of-sample fuzzy transform (FT), which extend their discrete counterparts to the continuous case. Through the out-of-sample FT, we introduce a coherent analysis of the discrete and continuous FTs, which is applied to extrapolate the behavior of the FT on new data and to achieve an accurate approximation of the continuous FT of signals on arbitrary data. To this end, we apply either an approximated approach, which considers the link between integral kernels and the spectrum of the corresponding Gram matrix, or an interpolation of the discrete kernel eigenfunctions with radial basis functions. In this setting, we show the generality of the proposed approach to the input data (e.g., graphs, 3-D domains) and signal reconstruction.


I. INTRODUCTION
D UE to the increasing availability of data, which is supported by ongoing technological advances in the acquisition, storage, and processing, several transformations [e.g., the Fourier transform, the Laplace transform, and the fuzzy transform (FT)] have been proposed to solve problems that spread from signal analysis to the solution of partial differential equations, from the analysis to the approximation of signals, and from fuzzy logic to fuzzy modelling.To deal with the definition of the FT on arbitrary data (see Section II), the data-driven FT [1] and the continuous FT [2] have been defined by applying the main concepts and results of spectral signal processing and manifold learning to the FT, e.g., dimensionality reduction [3] and kernels' eigenfunctions [4].The data-driven and continuous FTs provide a link between the fuzzy theory and previous work on diffusion kernels [5] and wavelets [5], [6], [7], [8], and geometric deep learning [9].
An input signal f : Ω → R is generally known at a set P of points and the corresponding FT Ff is computable only at P. As a result, we cannot estimate the value of Ff at a new point not belonging to P. Increasing the sampling of Ff , and consequently of the reconstructed signal F −1 (Ff ), will require adding new points in P and evaluating the corresponding function values, the FT, and its inverse.This last option is not generally feasible (e.g., if the signal values have been experimentally measured or their evaluation is time-consuming) and requires computing the FT and its inverse again.Since the FT of a continuous signal is still continuous, we expect its behavior to be accurately recovered at any point from a set of enough dense signal samples without further resampling the signal itself.The need to extrapolate the behavior of the FT Ff and of the reconstructed signal F −1 (Ff ) out of the set of input samples is further justified by the observation that most of the signals exist outside the input domain.For instance, we can measure the heat values at a set of points on a thin plate (i.e., a bounded 2-D domain) but recover the heat distribution on and around the plate.
For these reasons, we address the problem of defining the outof-sample membership functions and the resulting out-of-sample FT (see Section III), which extend their discrete counterparts to the continuous case.Through the out-of-sample FT, we introduce a coherent analysis of the discrete and continuous FTs, which extrapolate the behavior of the FT on new data and achieve an accurate approximation of the continuous FT of signals on arbitrary data (see Fig. 1).We also characterize the data-driven FT [1], [2] by representing the normalization factor in terms of the input filter and the area/volume of the input domain (see Section IV).As main properties of the out-of-sample extension of the FT, we discuss its linearity and continuity, self-adjointness and eigensystem, and convergence and fast computation.
To this end, we apply 1) an approximated approach (see Sections V and VI) that considers the link between integral kernels and the spectrum of the corresponding Gram matrix.The out-of-sample membership functions and the related FT are more robust to noisy data and tailored to address signal denoising and multiscale representations.Alternatively, we apply 2) an interpolation of the discrete kernel eigenfunctions with radial basis functions (RBFs), which is more accurate for "noise-free" data and useful for super-resolution and feature detection.Then, we characterize the relations between fuzzy theory and manifold learning, explicitly focusing on integral and spectral kernels.The out-of-sample membership functions and FT on large data are efficiently computed by approximating the L 2 (Ω) scalar product with a pseudoscalar product, which encodes the weights associated with graph edges or the area/volume of Voronoi regions for data represented as surface/volume meshes and preserves the main properties of the out-of-sample extension of the data-driven FT.Finally, we discuss different tests on the out-of-sample FT and signal reconstruction in the experimental part.

II. PREVIOUS WORK
We review the FT (see Section II-A), its extension to continuous FT [2] and the data-driven FT [1] (see Section II-B).

A. Fuzzy Tranform
Let us consider the space L 2 (Ω) of square integrable functions defined on a compact and connected domain Ω of R n , endowed with the L 2 (Ω) scalar product f, g 2 := Ω f (p)g(p)dp and the corresponding norm f 2 2 := Ω |f (p)| 2 dp.On the space C 0 (Ω) of continuous functions defined on Ω, we consider the L 2 (Ω) and the L ∞ (Ω)-norm f ∞ := max p∈Ω {|f (p)|}.Given a set P := {p i } n i=1 of points in Ω, a family of functions i=1 is a fuzzy partition of Ω if 1) A i is continuous, has its unique maximum 1 at p i ; 2) for all p ∈ Ω, n i=1 A i (p) = 1; for each i.Then, the FT [10], [11], [12], [13] of a function f : Ω → R is defined as the array Since the function f is known at a set of points The discrete FT is applied to recover an approximation f F,n of the function f underlying the set of values (f (q i )) s i=1 through the inverse FT [10], which is defined as f F,n (p) := FT as integral operator: Given a symmetric kernel A : Ω × Ω → [0, 1] (i.e., A(p, q) = A(q, p), p, q ∈ Ω), the continuous FT [1], [2] of f : Ω → R is defined as the continuous function Here, L K is the integral operator induced by the normalized kernel Noting that of the discrete FT.According to (2), any FT associated with the membership function A p : Ω → R is rewritten as an integral operator induced by the normalized kernel (3) with A(p, q) := A p (q). Viceversa, any symmetric kernel K : Ω × Ω → R can be considered as a membership function A p (q) : Ω → R, A p := K(p, q), and the corresponding FT is equal to (2).

III. OUT-OF-SAMPLE FT
In a general setting, an input signal f : Ω → R is known at a set P of points and the corresponding FT Ff is computable only at P. As a result, we cannot estimate the value of Ff at a new point not belonging to P. Increasing the sampling of Ff , and consequently of the reconstructed signal F −1 (Ff ), will require adding new points in P and evaluating the corresponding function values, the FT, and its inverse.This last option is not generally feasible (e.g., if the signal values have been experimentally measured or their evaluation is time-consuming) and requires computing the FT and its inverse again.
Since the FT of a continuous signal is still continuous, we expect that its behavior can be accurately recovered at any point from a set f of dense signal samples.To this end, we address the out-of-sample extension of the corresponding FT Ff =: F, i.e., the computation of a function E(Ff ) : The out-of-sample extension of the FT is a natural way to recover the continuous formulation of the FT from its discrete version.More precisely we study the relationship between the discrete and continuous FTs through the sampling operator R : We require that the out-of-sample operator is linear: i.e., E(αf + βg) = αEf + βEg ∀α, β ∀f , g.
Applying the out-of-sample operator to the set f := In this setting, the error between the data-driven FTs L K f and L K Ef is guided by the accuracy of the out-of-sample extension Ef of f .
In our examples, a signal f : Ω → R is represented through its colour-map and level-sets Then, we represent the out-of-sample extension Ef : R d → R through its colormap and isosurfaces Σ α := {p ∈ R d : Ef (p) = α}; analogously, for the out-of-sample extension E(Ff ) of the discrete generalized FT.The colour map varies the hue component of the hue-saturation-value colour model; the colours begin with red, pass through yellow, green, cyan, blue, and magenta, and return to red.a) Kernel-based out-of-sample FT: Selecting s seed points over n input points and recalling (3), we evaluate the n × s kernel matrix K := (K(p i , q j )) j=1,...,s i=1,...,n , K(p i , q j ) := In particular, K1 = 1, i.e., 1 is a singular vector of K and the corresponding singular value is 1.We compute the singular value decomposition [17]

Indeed, the functions
are the out-of-sample extensions of u i and v i , respectively.Equivalently, E(u i ) = u i and R(u i ) = u i .The out-of-sample kernel interpolates the entries of K, i.e., Furthermore, the functions (u i (•)) n i=1 are orthonormal with respect to the pointwise scalar product at P, i.e., as a consequence of the orthogonality of U.
Through the out-of-sample extensions of the left and righthand-side singular eigenvectors, we compute the out-of-sample extensions of the FT.Representing the discrete signal f = n i=1 f , u i 2 u i as a linear combination of the singular vectors Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Then, the out-of-sample spectral membership functions are and the out-of-sample FT is Equation ( 6) can also be derived from (2) as follows: From the relation K := D −1 A, we have that A = DK and the out-of-sample membership functions are b) Interpolating out-of-sample FT: Selecting s = n and p i = q i ∀i, the singular value decomposition of the symmetric kernel matrix K ∈ R n×n is replaced with its generalized eigendecomposition Kx i = λ i Dx i , with u i = v i = x i and σ i = λ i , x i Dx j = δ ij ∀i, j.In this case, the out-of-sample extension of the Laplacian eigenvectors minimizes the energy First, we compute its derivatives Recalling that the Laplacian eigenvectors are orthonormal In this case, the out-of-sample FT is (see Figs. 4 and 5) The approximation accuracy and convergence of E(L K f ) to L K f will be discussed in Section V. c) Experimental tests: For a given sampling P of Ω, we compute the generalized eigenvectors of (K, D), where K is the Gram matrix K := (K(p i , p j )) n i,j=1 , induced by the Gaussian kernel K(p, q) := exp( p − q 2 /σ), σ is the width parameter, and D is the diagonal matrix whose entries are the sum of the rows of K.The behavior of the colour map, the shape and the distribution of the level sets show the analogous behavior of the input and out-of-sample extension of the eigenfunctions, as confirmed by the low approximation error ∞ .The out-of-sample extension has a generally smoother behavior as a matter of its representation in terms of a smooth kernel (e.g., the Gaussian kernel).To analyze the properties of the proposed approach (see Figs. 4-7), we plot the singular values of the Gram matrix K induced by the selected kernel.We select three samples P 50 K , P 100 K , and P 150 K of a 3-D domain Ω, represented as a triangle mesh.The samplings P 100 K , P 150 K are achieved by splitting each triangle of P 50 K into four subtriangles by joining the midpoint of each edge; in this way, the new mesh has n V + 3n T vertices, where n V and n T are the number of vertices and triangles of the input mesh, respectively.We consider the discrete signal f 150 := (f (p i )) 150 K i=1 , achieved by sampling f : Ω → R at the point set P 150 K of the domain Ω at the highest resolution with 150 K samples.On these data sets, we evaluate the 1) extrapolation and 2) maximum errors between the input f and the out-of-sample extension Ef evaluated at P 50 K , P 100 K , and P 150 K .
For the results in Figs.4-7, the singular values rapidly decrease to zero, the order of magnitude of the pointwise error between f at P 150 K and (Ef 50 K )| P 150 K , (Ef 100 K )| P 150 K remains lower than 1%.The order of magnitude of the approximation error of the kernel-based approach remains lower than 10 −2 .According to (5) and assuming that the kernel matrix K is well conditioned, a low error in the approximation of the signal corresponds to an approximation of the corresponding generalized FT of the same order of magnitude.
Analogously to the previous tests, we compute the FT Ff 150 K = F 150 K := (F i ) 150 K i=1 of the input signal f : Ω → R on the sampling P 150 K of the domain Ω at the highest resolution with 150 K samples: we consider F 150 K as our ground-truth.We also compute the out-of-sample FTs EF 50 K , EF 100 K : R 3 → R of the signal at the lower resolutions of Ω with 50 K and 100 K samples and evaluate these two out-of-sample FTs at P 150 K .Then, we measure the pointwise and ∞ errors between the resulting FTs (EF 50 K )| P 150 K , (EF 150 K )| P 150 K and the groundtruth F 150 K , thus estimating the extrapolation capabilities of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the out-of-sample FT.In this case, the order of magnitude of the pointwise error remains lower than 1%.This behavior is consistent with the upper bound in (5) and the accuracy of the out-of-sample extension of the input signal.

IV. OUT-OF-SAMPLE DATA-DRIVEN FT
First (see Section IV-A), we express the data-driven FT in terms of the spectral filter and the area/volume of the input domain, thus avoiding the computation of integrals.Then, we introduce the out-of-sample extension of the data-driven FT (see Section IV-B).

A. Data-Driven FT: Unified Representation
We characterize the data-driven FTs [1], [2] by explicitly deriving the normalization factor in terms of the input filter and the area/volume of the input domain.Recalling that the Laplacian eigenfunctions are orthonormal and φ 0 = 1 is the eigenfunction associated with the null eigenvalue λ 0 , we get the relation Choosing the corresponding membership functions A p : Ω × Ω → R, centred at p and defined as A p (q) := K ϕ (p, q), we get that Let us introduce the normalized filter ϕ(s) := ϕ(s) ϕ(0)|Ω| , which is still positive and belongs to L 2 (R + ).Then, the spectral membership function centred at p is defined by the action of the data-driven FT on δ p , i.e., From ( 7), we rewrite (2) as The data-driven FT L K ϕ is linear and continuous , and positive-definite, according to the relation The data-driven FT is injective if and only if ϕ is strictly positive.In fact, is singular (i.e., ϕ(λ i ) = 0, for some i), then we consider its peudoinverse f = ϕ † (Δ)g.Discrete data-driven FT: Assuming that W is a weight matrix (e.g., the adjacency matrix), whose entry (i, j) is a strictly positive weight associated with the corresponding edge, the Laplacian matrix [18] is defined as L := D −1 L, where L := D − W and D is the diagonal matrix whose entries are the sum of the rows of W. The Laplacian matrix is D-adjoint with respect to the scalar product f , g D := f Dg, f := (f (p i )) n i=1 , i.e., Lf , g D = f , Lg D , and its spectral decomposition is LX = DXΛ, X DX = I, where X := [x 1 , . . ., x n ] is the eigenvectors' matrix and Λ is the diagonal matrix of the eigenvalues (λ i ) n i=1 .The discrete membership functions are represented in matrix form as and the discrete data-driven FT is where ϕ( L) is the filtered Laplacian matrix.

B. Out-of-Sample Data-Driven FT
For each Laplacian eigenvector x i := (x i (j)) n j=1 .we compute an out-of-sample extension j=1 , with RBFs Ψ(p, p i ) := ψ( p − p i 2 ), i = 1, . . ., n, induced by the generating function ψ : R → R. Imposing the interpolating conditions Ψ(i, j) := ψ( p i − p j 2 ) = Ψ(p i , p j ), we get the n × n linear system Ψα (i) = x i .The coefficient matrix is computed only once and applied to evaluate all the out-of-sample extensions of the Laplacian eigenvectors.In particular, E( We also report statistics on the singular values' distribution, approximation, and pointwise errors. of the Laplacian eigenvectors as f = n i=1 f , x i D x i , the function is the out-of-sample extension of f ; in fact Once we have computed the out-of-sample extension of the Laplacian eigenvectors, the spectral membership function at p j is evaluated as K ϕ(p j , q) := n i=1 ϕ(λ i )ψ i (p j )ψ i (q), p ∈ Ω, and the corresponding out-of-sample spectral FT is (see Figs. 8-11) We also report statistics on the singular values' distribution, approximation, and pointwise errors.
which is analogous to (4).In a similar way where the last integral is easily computed as the generating function ψ is 1-D.
Computation: For the out-of-sample extension with RBFs, the evaluation of (10) requires computing the component through quadrature rules with RBFs.Increasing the sampling density of Ω (see Fig. 13), the behavior of the out-of-sample extensions of the input signal and the corresponding generalized FT remain almost unchanged and coherent in terms of the variation of the colourmap and distribution/shape of the iso-surfaces.Furthermore, the maximum variation of Ef , Ff , and E(Ff ) on the different samplings is lower than 10 −2 .

V. APPROXIMATED OUT-OF-SAMPLE DATA-DRIVEN FT
Since the out-of-sample extensions (ψ i ) n i=1 are not orthonormal, we introduce a pseudoscalar product that approximates the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.L 2 (Ω) product and makes the out-of-sample eigenfunctions orthonormal (see Section V-A).This pseudoscalar product also induces an approximation of the out-of-sample FT that is independent of the evaluation of the integral Ω f (p)ψ( p − p i 2 )dp and converges to (10), as the sampling density of Ω increases.The approximated out-of-sample data-driven FT still satisfies the main properties of the FT, such as linearity and continuity, self-adjointness and eigensystem, surjectivity, convergence, and fast computation (see Section V-C).

A. Pseudoscalar Product
Given f, g ∈ C 0 (Ω) and P := {p i } n i=1 samples of Ω, let us consider the pseudoscalar product f, g := f , g D = f Dg (11) where f := (f (p i )) n i=1 , g := (g(p i )) n i=1 , and D is a positive definite matrix (e.g., D := I or the mass matrix in Section IV-A).Since • D is a norm, the pseudonorm f ≥ 0 and f = 0 implies only that f (p i ) = 0, i = 1, . . ., n; 4) is distributive f, (g + h) = f, g + f, h and commutative: f, g = g, f ; 5) is convergent: increasing the sampling density of Ω, we get that the pseudoinner product provides an increasingly more accurate approximation of the L 2 (Ω) scalar product.Furthermore, the function f approx := n i=1 f , x i D φ i is the best least-squares approximation of f in the space S := span{φ i } n i=1 with respect to the pseudonorm • ; in fact, the minimum of the error is achieved for α i := f , x i D .Indeed, among all the interpolating functions of f the function f approx is unique and minimizes the • norm.
Comparison between generalized and out-of-sample extension of the FT: Let us now estimate the difference between the generalized FT Ff and the out-of-sample extension of the discrete FT E(L K f ).From the definition of the generalized FT and the spectral representation of the our-of-sample kernel, we get that which converges to zeros as the sampling density increases; in fact, λ i → 0 and d(i) f , x i 2 → f, φ i 2 , as i → +∞.

B. Approximated out-of-sample data-driven FT
From (11), we have that Recalling (4), we define the approximated out-of-sample FT L approx Then, ( 13) is efficiently computed as the term f, φ i 2 in ( 10) is now replaced by f , x i D .
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C. Properties 1) Linearity and Continuity:
The approximated out-of-sample FT is linear L approx K ϕ (αf + βg) = αL approx K ϕ f + βL approx K ϕ g, and "pseudo" continuous, according to the following uppper bound: is linear with respect to the input filter, i.e., 2) Self-Adjointness and Spectrum: The approximated outof-sample FT is self-adjoint with respect to the pseudoscalar product, i.e., In particular, L approx K ϕ 1 = 0.
3) Convergence: Let us now estimate the approximation error between the out-of-sample FT and its approximation which converges to zero as x i , f D converges to φ i , f 2 , as the sampling density of Ω increases.The same result applies when considering the norm • .4) Fast Computation: According to the upper bound a good approximation ϕ 2 of ϕ 1 guarantees that L approx .For instance, let p be a polynomial (or rational polynomial) approximation of ϕ such that r ∞ := ϕ − p ∞ .Then, Indeed, the filter ϕ is approximated with a polynomial (or a rational polynomial) p, which allows us to evaluate (L approx ϕ f )(p) without computing the Laplacian spectrum, which is generally unfeasible in terms of computational cost and storage overhead.

5) Restriction and Out-of-Sample Extensions:
Applying the linearity of the out-of-sample extension and restriction operators, and the relations E(x i ) = φ i , R(φ i ) = x i , we have that L K ϕ is the out-of-sample extension of the discrete spectral operatir L ϕ, i.e., L K ϕ f = E(L ϕf ).In fact, (L K ϕ f )(p i ) = (L ϕf )(i), i = 1, . . ., n.According to the previous relations, the pseudoscalar product and the approximated out-of-sample FT provide a coherent approximation of the FT.Similarly, we derive the relations R(L K ϕ f ) = L ϕf and L K ϕ (Ef ) = L K ϕ f .6) Experimental Tests: Given a discrete function f (see Figs. 12 and 13), we compute its out-of-sample extension according to (9), its discrete FT F := L K f , and the corresponding out-of-sample extension E(F).The behavior of the discrete FT F is represented through its colour map and level sets; the behavior of the out-of-sample extension E(F) : R d → R is represented through its iso-surface Σ α := {p ∈ R d : EF(p) = α}.We notice the consistency between the behavior of F and EF, where each level set γ α on Ω corresponds to an iso-surface Σ α of E(F).

VI. CONCLUSION AND FUTURE WORK
This article has presented the definition and computation of the out-of-sample membership functions and the resulting out-of-sample FT, which extend their discrete counterparts to the continuous case.Through the out-of-sample FT, we have achieved a coherent analysis of the discrete and continuous FTs, which is applied to extrapolate the behavior of the FT on new data and to achieve an accurate approximation of the continuous FT of signals on arbitrary data.As main future work, we plan to investigate 1) the class of functions used for the out-of-sample FT and 2) the definition of the generating kernel from the input data better to adapt the out-of-sample approximation of the input signal and of the FT to the data itself, e.g., generating these functions and kernels from the input data through learning and manifold learning.

Fig. 1 .
Fig. 1.Given a function f : Ω → R, the scheme summarized the definition of the out-of-sample extensions Ef , E(Ff ) of f and its FT Ff .
(Ch. 2) UKV = Γ of the kernel matrix, with orthogonality conditions U U = I and V V = I, U ∈ R n×n , V ∈ R s×s .Here, Γ ∈ R n×s has p := min{n, s} nonnull diagonal entries.According to the relations KV = UΓ, K U = VΓ , we rewrite the left and right eigenvectors as (Figs. 2 and 3, 1st column)

Fig. 2 .
Fig. 2. Color-map and level-sets of the input singular vectors and Laplacian eigenvectors.Iso-surfaces of the kernel-based (see Section IV) out-of-sample extension of the singular vectors and interpolating out-of-sample extension of the kernel eigenvectors (see Section IV-B).

Fig. 3 .
Fig. 3. Color-map and level-sets of the input singular vectors u i and Laplacian eigenvectors φ i .Iso-surfaces of the kernel-based (see Section IV) out-of-sample extension Eu i , Eφ i of the singular vectors and interpolating out-of-sample extension of the kernel eigenvectors (see Section IV-B).

Fig. 4 .Fig. 5 .Fig. 6 .
Fig. 4. Statistics of the kernel-based out-of-sample spectral FT on (a) an input and up-sampled domain Ω, represented by a point set P and its resampling P split .See also Fig. 5.

Fig. 7 .
Fig.7.With reference to Fig.6, colour map and level sets of the input sognal f and its out-of-sample extension Ef : R d → R and its restriction to P split , Ef | P split : P split → R, on an upsampling P split of Ω.

Fig. 8 .
Fig. 8.Given a signal f : Ω → R, we compute (a) its FT Ff , (b) its FT on an up-sampling P split of Ω, and (c) the kernel-based out-of-sample FT E(Ff ).We also report statistics on the singular values' distribution, approximation, and pointwise errors.

Fig. 9 .
Fig. 9. (a) Color map and level sets of an input signal f : Ω → R and isosurfaces of its out-of-sample extension Ef : R d → R, (b) colour map and level sets of the FT Ff : Ω → R and iso-surfaces of its kernel-based out-of-sample extension E(Ff ) : R d → R.

Fig. 10 .
Fig. 10.Given a signal f : Ω → R, we compute (a) its FT Ff , (b) its FT on an up-sampling P split of Ω, and (c) the kernel-based out-of-sample FT E(Ff ).We also report statistics on the singular values' distribution, approximation, and pointwise errors.

Fig. 11 .
Fig. 11.(a, b) Statistics on the singular values.distribution, approximation, and pointwise errors.(c,e) Color map and level sets of an input signal f : Ω → R and iso-surfaces of its kernel-based out-of-sample extension Ef : R d → R, (d,f) colour map and level sets of the FT Ff and iso-surfaces of its out-of-sample extension E(Ff ) : R d → R.

Fig. 12 .
Fig. 12. Kernel-based out-of-sample extension Ef : R d → R of the input signal and its kernel-based out-of-sample extension E(Ff ) : R d → R of the FT on 3-D domains.

Fig. 13 .
Fig. 13.Out-of-sample extension Ef : R d → R of the input signal and its out-of-sample extension E(Ff ) : R d → R of the FT on a 3-D domain with an increasing sampling density.

K ϕ 2
is a good approximation of L approx K ϕ 1