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Photonic neural networks offer a promising alternative to traditional electronic systems for machine learning accelerators due to their low latency and energy efficiency. However, the challenge of implementing the backpropagation algorithm during training has limited their development. To address this, alternative machine learning schemes, such as extreme learning machines (ELMs), have been proposed. ELMs use a random hidden layer to increase the feature space dimensionality, requiring only the output layer to be trained through linear regression, thus reducing training complexity. Here, we experimentally demonstrate a programmable photonic extreme learning machine (PPELM) using a hexagonal waveguide mesh, and which enables to program directly on chip the input feature vector and the random hidden layer. Our system also permits to apply the nonlinearity directly on-chip by using the systems integrated photodetecting elements. Using the PPELM we solved successfully three different complex classification tasks. Additioanlly, we also propose and demonstrate two techniques to increase the accuracy of the models and reduce their variability using an evolutionary algorithm and a wavelength division multiplexing approach, obtaining excellent performance. Our results show that programmable photonic processors may become a feasible way to train competitive machine learning models on a versatile and compact platform.

Photonic integrated circuits play an important role in the field of optical computing, promising faster and more energy-efficient operations compared to their digital counterparts. This advantage stems from the inherent suitability of optical signals to carry out matrix multiplication. However, even deterministic phenomena such as thermal crosstalk make precise programming of photonic chips a challenging task. Here, we train and experimentally evaluate three models incorporating varying degrees of physics intuition to predict the effect of thermal crosstalk in different locations of an integrated programmable photonic mesh. We quantify the effect of thermal crosstalk by the resonance wavelength shift in the power spectrum of a microring resonator implemented in the chip, achieving modelling errors <0.5 pm. We experimentally validate the models through compensation of the crosstalk-induced wavelength shift. Finally, we evaluate the generalization capabilities of one of the models by employing it to predict and compensate for the effect of thermal crosstalk for parts of the chip it was not trained on, revealing root-mean-square-errors of <2.0 pm., Comment: 10 pages, 10 figures. The paper has been submitted to the Journal of Lightwave Technology, a special issue on Photonic Computing, as an invited paper. It is under review

Data center networks are experiencing unprecedented exponential growth, mostly driven by the continuous computing demands in machine learning and artificial intelligence algorithms. Within this realm, optical networking offers numerous advantages, including low latency, energy efficiency, and bandwidth transparency, positioning it as a compelling alternative to its electronic counterparts. In this work, we showcase a range of software-defined optical networking applications deployed on a general-purpose programmable integrated photonic processor. Leveraging graph-based theory, we experimentally demonstrate dynamic optical interconnects, circuit switching, and multicasting on the same photonic platform, yielding remarkable results in terms of crosstalk and reconfiguration speed. Our approach harnesses the benefits of reconfigurability and reliability, paving the way for a new generation of high-performance optical devices tailored for data center and computing clusters.

Recommender systems are intrinsically tied to a reliability/coverage dilemma: The more reliable we desire the forecasts, the more conservative the decision will be and thus, the fewer items will be recommended. This causes a detriment to the predictive capability of the system, as it is only able to estimate potential interest in items for which there is a consensus in their evaluation, rather than being able to estimate potential interest in any item. In this paper, we propose the inclusion of a new term in the learning process of matrix factorization-based recommender systems, called recklessness, that takes into account the variance of the output probability distribution of the predicted ratings. In this way, gauging this recklessness measure we can force more spiky output distribution, enabling the control of the risk level desired when making decisions about the reliability of a prediction. Experimental results demonstrate that recklessness not only allows for risk regulation but also improves the quantity and quality of predictions provided by the recommender system., Comment: 15 pages, 4 figures, 2 tables

A variety of topological Hamiltonians have been demonstrated in photonic platforms, leading to fundamental discoveries and enhanced robustness in applications such as lasing, sensing, and quantum technologies. To date, each topological photonic platform implements a specific type of Hamiltonian with inexistent or limited reconfigurability. Here, we propose and demonstrate different topological models by using the same reprogrammable integrated photonics platform, consisting of a hexagonal mesh of silicon Mach-Zehnder interferometers with phase-shifters. We specifically demonstrate a one-dimensional Su-Schrieffer-Heeger Hamiltonian supporting a localized topological edge mode and a higher-order topological insulator based on a two-dimensional breathing Kagome Hamiltonian with three corner states. These results highlight a nearly universal platform for topological models that may fast-track research progress toward applications of topological photonics and other coupled systems.

Unsupervised machine learning lacks ground truth by definition. This poses a major difficulty when designing metrics to evaluate the performance of such algorithms. In sharp contrast with supervised learning, for which plenty of quality metrics have been studied in the literature, in the field of dimensionality reduction only a few over-simplistic metrics has been proposed. In this work, we aim to introduce the first highly non-trivial dimensionality reduction performance metric. This metric is based on the sectional curvature behaviour arising from Riemannian geometry. To test its feasibility, this metric has been used to evaluate the performance of the most commonly used dimension reduction algorithms in the state of the art. Furthermore, to make the evaluation of the algorithms robust and representative, using curvature properties of planar curves, a new parameterized problem instance generator has been constructed in the form of a function generator. Experimental results are consistent with what could be expected based on the design and characteristics of the evaluated algorithms and the features of the data instances used to feed the method., Comment: 16 pages, 4 figures, submitted to IEEE Transactions on Pattern Analysis and Machine Intelligence

We are concerned with bilinear estimates and uniqueness of mild solutions for the Navier-Stokes equations in critical spaces. For that, we construct general settings in which estimates for the bilinear term of the mild formulation hold true without using auxiliary norms such as Kato time-weighted ones. We first obtain necessary conditions in abstract critical spaces and then consider further structures to obtain the estimates in general classes of Besov, Morrey and Besov-Morrey spaces based on Banach spaces. Examples of applications are provided in different spaces as well as for other PDEs. In particular, as far as we know, the bilinear estimate and the uniqueness property obtained in the framework of Besov-weak-Herz spaces are not available in the existing literature. The proofs are mainly based on characterizations and estimates on the corresponding predual spaces., Comment: 39 pages

We consider real hypersurfaces $M$ in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull constant $k$ and any symmetric tensor field of type (1,1) $L$ on $M$ we can define two tensor fields of type (1,2) on $M$, $L_F^{(k)}$ and $L_T^{(k)}$, related to both connections. We study the behaviour of the structure operator $\phi$ with respect to such tensor fields in the particular case of $L=A$, the shape operator of $M$, and obtain some new characterizations of ruled real hypersurfaces in complex projective space.

The linearization of a microwave photonic link based on a dual-parallel Mach-Zehnder modulator is theoretically described and experimentally demonstrated. Up to four different radio frequency tones are considered in the study, which allow us to provide a complete mathematical description of all third-order distortion terms that arise at the photodetector. Simulations show that a complete linearization is obtained by properly tuning the DC bias voltages and processing the optical carrier and. As a result, a suppression of 17 dBm is experimentally obtained for the third-order distortion terms, as well as a SDFR improvement of 3 dB. The proposed linearization method enables the simultaneous modulation of four different signals without the need of additional radio frequency components, which is desirable to its implementation in integrated optics and makes it suitable for several applications in microwave photonics.

The almost contact metric structure that we have on a real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ allows us to define, for any nonnull real number $k$, the $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. Associated to this connection we have Cho and torsion operators, $F_X^{(k)}$ and $T_X^{(k)}$, respectively, for any vector field $X$ tangent to $M$. From them and for any symmetric operator $B$ on $M$ we can consider two tensor fields of type (1,2) on $M$ that we will denote by $B_F^{(k)}$ and $B_T^{(k)}$, respectively. We will classify real hypersurfaces $M$ in $Q^m$ for which any of those tensors identically vanishes, in the particular case of $B$ being the structure Lie operator $L_{\xi}$ on $M$., Comment: 9 pages

This paper is concerned with the fractional Keller-Segel system in the temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag-Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller-Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition of the initial data.

Digital electronics is a technological cornerstone in our modern society which has covered the increasing demand in computing power during the last decades thanks to a periodic doubling of transistor density and power efficiency in integrated circuits. Currently, such scaling laws are reaching their fundamental limits, leading to the emergence of a large gamut of applications that cannot be supported by digital electronics, specifically, those that involve real time analog multi-data processing, e.g., medical diagnostic imaging, robotic control and remote sensing, among others. In this scenario, an analog computing approach implemented in a real-time reconfigurable non-electronic hardware such as programmable integrated photonics (PIP) can be more efficient than digital electronics to perform these emerging applications. However, actual analog computing models such as quantum and neuromorphic computation were not conceived to extract the benefits of PIP technology (and integrated photonics in general). Here, we present the foundations of a new computation theory, termed Analog Programmable-Photonic Computation (APC), explicitly designed to unleash the full potential of PIP. Interestingly, APC enables overcoming basic theoretical and technological limitations of existing computational models, can be implemented in other technologies (e.g. in electronics, acoustics or using metamaterials) and, consequently, exhibits the potential to spark a ground-breaking impact on our information society.