The Moment-SOS hierarchy and the Christoffel-Darboux kernel

We consider the global minimization of a polynomial on a compact set B. We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of L 2 (B,μ) where μ is an arbitrary reference measure whose support is exactly B. The resulting polynomial is a certain density (with respect to μ) of some signed measure on B. When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel-Darboux (CD) kernel and the Christoffel function associated with μ. In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).

Phase transitions for spatially extended pinning

We consider a directed polymer of length N interacting with a linear interface. The monomers carry i.i.d. random charges $$(\omega _i)_{i=1}^N$$ ( ω i ) i = 1 N taking values in $${\mathbb {R}}$$ R with mean zero and variance one. Each monomer i contributes an energy $$(\beta \omega _i-h)\varphi (S_i)$$ ( β ω i - h ) φ ( S i ) to the interaction Hamiltonian, where $$S_i \in {\mathbb {Z}}$$ S i ∈ Z is the height of monomer i with respect to the interface, $$\varphi :\,{\mathbb {Z}}\rightarrow [0,\infty )$$ φ : Z → [ 0 , ∞ ) is the interaction potential, $$\beta \in [0,\infty )$$ β ∈ [ 0 , ∞ ) is the inverse temperature, and $$h \in {\mathbb {R}}$$ h ∈ R is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on $${\mathbb {Z}}$$ Z . We study both the quenched and the annealed free energy per monomer in the limit as $$N\rightarrow \infty $$ N → ∞ . We show that each exhibits a phase transition along a critical curve in the $$(\beta ,h)$$ ( β , h ) -plane, separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We derive variational formulas for the critical curves and we obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we derive the scaling limit of the annealed free energy as $$\beta ,h \downarrow 0$$ β , h ↓ 0 in three different regimes for the tail exponent of $$\varphi $$ φ .

Sobolev–Kantorovich inequalities under CD(0,∞) condition

We refine and generalize several interpolation inequalities bounding the [Formula: see text] norm of a probability density with respect to the reference measure [Formula: see text] by its Sobolev norm and the Kantorovich distance to [Formula: see text] on a smooth weighted Riemannian manifold satisfying [Formula: see text] condition.

Changing the Geometry of Representations: α-Embeddings for NLP Tasks

Word embeddings based on a conditional model are commonly used in Natural Language Processing (NLP) tasks to embed the words of a dictionary in a low dimensional linear space. Their computation is based on the maximization of the likelihood of a conditional probability distribution for each word of the dictionary. These distributions form a Riemannian statistical manifold, where word embeddings can be interpreted as vectors in the tangent space of a specific reference measure on the manifold. A novel family of word embeddings, called α-embeddings have been recently introduced as deriving from the geometrical deformation of the simplex of probabilities through a parameter α, using notions from Information Geometry. After introducing the α-embeddings, we show how the deformation of the simplex, controlled by α, provides an extra handle to increase the performances of several intrinsic and extrinsic tasks in NLP. We test the α-embeddings on different tasks with models of increasing complexity, showing that the advantages associated with the use of α-embeddings are present also for models with a large number of parameters. Finally, we show that tuning α allows for higher performances compared to the use of larger models in which additionally a transformation of the embeddings is learned during training, as experimentally verified in attention models.

An Approach towards Motion-Tolerant PPG-Based Algorithm for Real-Time Heart Rate Monitoring of Moving Pigs

Animal welfare remains a very important issue in the livestock sector, but monitoring animal welfare in an objective and continuous way remains a serious challenge. Monitoring animal welfare, based upon physiological measurements instead of the audio–visual scoring of behaviour, would be a step forward. One of the obvious physiological signals related to welfare and stress is heart rate. The objective of this research was to measure heart rate (beat per minutes) in pigs with technology that soon will be affordable. Affordable heart rate monitoring is done today at large scale on humans using the Photo Plethysmography (PPG) technology. We used PPG sensors on a pig′s body to test whether it allows the retrieval of a reliable heart rate signal. A continuous wavelet transform (CWT)-based algorithm is developed to decouple the cardiac pulse waves from the pig. Three different wavelets, namely second, fourth and sixth order Derivative of Gaussian (DOG), are tested. We show the results of the developed PPG-based algorithm, against electrocardiograms (ECG) as a reference measure for heart rate, and this for an anaesthetised versus a non-anaesthetised animal. We tested three different anatomical body positions (ear, leg and tail) and give results for each body position of the sensor. In summary, it can be concluded that the agreement between the PPG-based heart rate technique and the reference sensor is between 91% and 95%. In this paper, we showed the potential of using the PPG-based technology to assess the pig′s heart rate.

An Approach Towards Motion-Tolerant PPG-Based Algorithm for Real-Time Heart Rate Monitoring of Moving Pigs

Animal welfare remains a very important issue in the livestock sector but monitoring animal welfare in an objective and continuous way remains a serious challenge. Monitoring animal welfare based upon physiological measurements instead of audio-visual scoring of behaviour would be a step forward. One of the obvious physiological signals related to welfare and stress is heart rate. The objective of this research was to measure heart rate (beat per minutes) on pigs with technology that soon will be affordable. Affordable heart rate monitoring is done today at large scale on humans using the Photo Plethysmography (PPG) technology. We used PPG sensors on pig’s body to test whether it allows getting reliable heart rate signal. A continuous wavelet transform (CWT)-based algorithm is developed to decouple the cardiac pulse waves from the pig. Three different wavelets, namely 2nd, 4th and 6th order Derivative of Gaussian (DOG, are tested. We show results of the developed PPG-based algorithm against electrocardiograms (ECG) as a reference measure for heart rate and this for an anesthetized versus a non-anesthetised animal. We tested three different anatomical body positions (ear, leg and tail) and give results for each body position of the sensor. In summary, it can be concluded that the agreement between PPG-based heart rate technique and reference sensor goes from 91 to 95 percentage. In this paper we showed the potential of using the PPG-based technology to assess pig’s hear rate.

Conditional Rényi Entropy and the Relationships between Rényi Capacities

The analogues of Arimoto’s definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored.

System of metrological assurance of traceability of measurements of mass geometry characteristics

A system of metrological support is proposed that allows testing for type approval, verification and calibration of stands for measuring mass, coordinates of mass center and moments of inertia. The mathematical model of a special standard has been developed. It is based on the analytical principles for the determination of the mass center coordinates and inertia moments of homogeneous bodies with a regular geometric shape. The standard consists of a set of modules and fasteners of a special shape. Each module is a reference measure of both mass, center of mass coordinates, and moments of inertia, and can be used either separately or in a set with other modules. A scheme for transferring units of values from state primary standards of mass and length to stands using special standards has been developed. A method for calibration of special standards has been developed. It includes indirect measurements of the coordinates of the center of mass and moments of inertia based on the results of direct measurements of the mass and geometric dimensions of the standard modules’ elements, as well as measurements of the form deviations and deviations of position of the standard modules’ surfaces and static balancing of the standard. Technical requirements for special standards have been developed to minimize methodological measurement uncertainties when reproducing units of values by the standard. A line of special standards was created, their certification and approval in Rosstandart was carried out. Stand verification methods using special standards have been developed and approved. The type approval of two stands was carried out using special standards. The decision to grant a patent for the invention “Method for determining the error of the stand for measuring the characteristics of the mass geometry of products and a device for its implementation” was received.