Research

Here is an overview on my research topics, including relevant applications and some related papers.
Visit my ResearchGate webpage and the CV for further details and resources!

List of topics (in an approximate chronological order):

Filtered polynomial approximation at Lissajous nodes

Lissajous curves are generated by superimposing two perpendicular oscillations, of different angular frequency, in x and y directions. A particular family of Lissajous curves turns out to be interesting from an approximation viewpoint, as bivariate polynomial interpolation at the related points is stable and effective.

Two examples of Lissajous curve and related points, which correspond to self-intersections and tangent points with the square.

Left: a degenerate Lissajous curve, which in this case generates the so-called Padua points.

Right: a non-degenerate Lissajous curve. Note that its period is two times longer than the degenerate case, with many more nodes generated.

Polynomial approximation methods built along Lissajous curves provide a fast image reconstruction in Magnetic Particle Imaging. However, a Gibbs phenomenon is likely to occur in the reconstructed image, because of the presence of edges in the underlying function. We studied  an adaptive spectral filtering process for the resolution of this phenomenon and for an improved approximation of the underlying image. The adaptive spectral filter takes into account the distance with respect to the closest edge of the phantom. Such discontinuities are possibly localized by means of an edge detector.

A simulated MPI reconstruction. The phantom (left). Distorted  reconstruction (center). Filtered reconstruction with edges highlighted in green (right).
References:

Mapped bases approaches (... That is, fake nodes!)

The position of the interpolation data sites has large influence on both the stability and the accuracy of the reconstruction process. Even in the simple univariate polynomial interpolation setting, it is well known that considering equidistant nodes might cause the appearance of the Runge phenomenon. Moreover, the Gibbs phenomenon almost inevitable when dealing with a discontinuous underlying function.
The approach that we investigated can be intended equivalently as a mapped bases or a fake nodes approach. With the first nomenclature, we remark that we are indeed constructing the interpolant in a mapped space. With the second we put a focus on the construction of this mapping, which is chosen in such a way that the original set of nodes is mapped onto a certain set of fake nodes. The fake nodes are well-behaved for a classical interpolation process with the original bases functions, and therefore the whole process guarantees the benefits of a good interpolation design without paying the price of resampling the underlying function. 


The fake nodes approach in univariate mapped polynomial interpolation.

Left: equidistant nodes are mapped on Chebyshev-Lobatto nodes (we denoted it as the S-Runge scheme).

Right: equidistant nodes are partially shifted to overcome the effects of the Gibbs phenomenon (we named it the S-Gibbs algorithm).

This approach has been then extended to a more general framework, including multivariate polynomial interpolation and rational barycentric interpolation, and to the quadrature context.

References:

(β,γ)-Chebyshev functions and points

While applying the mentioned fake nodes approach, we noticed that mapping certain sets of nodes that are equidistant in some subinterval of [-1,1] produced well-behaved interpolation designs, which we then formalized as a generalization of the classical Chebyshev nodes of the interval. Therefore, we studied this new family of nodes and related functions, which depends on the value of two parameters. We then exploited the new findings in the construction of a mapping that can lead to the avoidance of both Runge's and Gibbs phenomenon in univariate mapped polynomial interpolation.

Examples of (β,γ)-Chebyshev functions (solid lines) and points (blue dots and red crosses) for different values of the parameters. They can be similar to the classical Chebyshev ones, or be localized in subintervals of [-1,1].
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Variably Scaled (Discontinuous) Kernels

Variably Scaled Kernels (VSKs) have been introduced in 2015 to overcome instability issues in kernel-based interpolation with radial kernels (i.e., RBF interpolation). This strategy consists of augmenting the original space by adding a further coordinate to the elements of the domain. Then, after applying a classical RBF interpolation process in the augmented space, we can forget the added coordinate and project back to the original domain.
Of course, the effeciveness of this strategy depends on the way we choose the added coordinate, which is ruled by the so-called scaling function. We studied Variably Scaled Discontinuous Kernels (VSDKs), which are generated by selecting a discontinuous scaling function. If the jumps of the scaling function replicates the ones of the underlying function to be reconstructed, exactly or approximately, the interpolation process with VSDKs outperforms classical RBF interpolation.

Left: a classical radial basis function centered at the origin.

Right: by means of a discontinuous scaling function, we can realize a basis function that presents jumps. Since this basis function is the projection of a classical radial function from an augmented space, the interpolation process is well-posed, being equivalent to a classical one in a different space.
VSDKs for MPI, from left to right: the samples at Lissajous nodes, the original reconstruction, the scaling function for constructing the discontinuous kernel, the final reconstruction where the Gibbs phenomenon has been reduced.

Besides the discontinuous case, we explored the idea of VSKs in the framework of moving least squares and in combination with the fake nodes approach. Moreover, in the machine learning classification context VSKs can be interpreted as a stacking technique. In this direction, we employed VSKs to encode information from auxiliary classifiers into the support vector machines scheme, and in persistent homology applications.




In the field of solar imaging, the original data sites produced by STIX are clustered, which is not a desirable property from a RBF interpolation perspective.

By combining the fake nodes approach with VSKs, we can interpolate at a better distributed interpolation design.
References:

Improved cross validation algorithms for meshfree interpolation and collocation

In RBF interpolation, the basis functions usually depend on a real-valued positive shape parameter that determines the "level of flatness" of the radial function. Tuning this parameter is an important issue, which can be addressed by using a cross validation scheme. However, a standard implementation of k-fold cross validation is really demanding from a computational viewpoint. Starting from a Leave-One-Out cross validation algorithm proposed by S. Rippa in 1999, we designed and then applied a faster implementation of k-fold cross validation for both kernel-based interpolation and collocation settings.


References:

Efficient local-to-global support vector machines

Support Vector Machines (SVMs) are very popular in the context of machine learning supervised classification. However, the computational cost required for their employment might be quite demanding in presence of large datasets. Inspired by the approximation framework, we adapted the Partition of Unity (PU) method to work in SVM classification. As a result, we can construct a Local-to-Global SVM (LGSVM) classifier that is built upon many local SVMs models, which are assembled together by taking advantage of the corresponding overlapping subdomains that characterize the PU approach. LGSVMs are competitive with classical SVMs, but the related computational cost is severely dimished.



Starting from a 2D dataset (left), we can construct a PU covering by considering subdomains that are balls in the Euclidean metric (center) or in the cosine semi-metric (right).
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Score oriented loss functions for deep learning methods (with applications!)

In supervised deep learning classification, loss minimization and classification score maximization are intertwined aspects. However, a score can not be considered as a loss function, because it usually depends on a threshold that determines the entries of the confusion matrix. As a consequence, the score is typically discontinuous with respect to the weights of the network. Our approach consists in treating such threshold not as a fixed value, but as a random variable with a certain probability density function (pdf). Then, we can construct a Score-Oriented Loss (SOL) function by exploiting an expected score defined on the entries of probabilistic confusion matrices, where the probabilistic threshold is taken into account.
As a result, the SOL provides an automatic optimization of the chosen classification score directly in the training phase. Furthermore, the pdf associated with the threshold significantly impacts the outcome of the score maximization process, because it heavily influences the a posteriori best threshold value that maximizes the score. In the figure below, in blue we display the pdf chosen for the threshold, while in red we highlight the threshold values that empirically maximize the considered classification score, which are obtained by means of a posteriori tuning over one hundred random repetitions. The related mean and standard deviation are outlined with the red cross and bar, respectively.

We applied SOLs in the field of space weather forecasting, where we considered HMI video data to predict the occurrence and magnitude of a solar flare in the next 24 hours.

Left: an image from HMI, onboard SDO.

Right: two videos obtained from HMI data. Predicting the magnitude of a occurring flare (in this example, above C class or not) is a non-trivial task.
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Physics-driven machine learning for predicting CMEs' travel time

Another important task in space weather is to predict the travel time of Coronal Mass Ejections (CMEs) from Sun to Earth. We studied a physics-driven machine learning approach where the widely-used drag-based  physical model is exploited in the definition of the loss function of the neural network model. In our experiments, we noticed that including such a physical information leads to more robust and accurate results with respect to a fully data-driven approach.

Left: a visualization of a CME.

Right: the architecture that we considered in our paper, it consists of a cascade of two feed-forward neural network where physics information is encoded.
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SSIM-convergence of image interpolation schemes 

The Structural Similarity Index Measure (SSIM) has been introduced in order to assess the preceived similarity between two images. In literature, many examples where SSIM and mean-squared error (or L2-error) behave differently in judging the similitude between images have been provided. In our works, we analyzed the relationship between SSIM and L2-error, showing that they are actually equivalent in some circumstances. This was done by considering a continuous extension of the natively discrete SSIM, and by carrying out a formal convergence analysis of different image interpolation methods.

References:

Data-driven optimized kernel machines for data approximation

As previously mentioned, RBF approximation models usually depends on a real-valued positive shape parameter, whose tuning is for sure a non-trivial issue. We studied a more general approach where the single shape parameter is substituted by a full matrix of parameters, that allows an enhanced capability of the basis to adapt to the interpolation design. In order to set this set of hyperparameters, we leverage tools from the machine learning framework, as stochastic gradient descent optimization. Finally, a greedy procedure with the so-optimized kernel allows the construction of a model that is definitelytailored with respect to the data at disposal.




The proposed model can be considered as a two-layered kernel machine, where the entries of the matrix A that acts on the data need to be learned.
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Modeling water distribution systems via the graph p-Laplacian operator

The graph p-Laplacian operator is the non-linear extension of the well-known Laplacian operator on graphs. Due to his properties, it represents a valuable tool to construct surrogate models of water distribution systems. However, a good tuning of the p parameter and of the edges weight is required to obtain an effective model. Performing this tuning by starting from data sampled at edges and nodes represents a non-linear inverse problem on possible large graph structures.


References:
... Soon!