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Quantifying uncertainty using confidence regions is a central goal of statistical inference. Despite this, methodologies for confidence bands in functional data analysis are still underdeveloped compared to estimation and hypothesis testing. In this work, we present a new methodology for constructing simultaneous confidence bands for functional parameter estimates. Our bands possess a number of positive qualities: (1) they are not based on resampling and thus are fast to compute, (2) they are constructed under the fairness constraint of balanced false positive rates across partitions of the bands' domain which facilitates the typical global, but also novel local interpretations, and (3) they do not require an estimate of the full covariance function and thus can be used in the case of fragmentary functional data. Simulations show the excellent finite-sample behaviour of our bands in comparison to existing alternatives. The practical use of our bands is demonstrated in two case studies on sports biomechanics and fragmentary growth curves. [ABSTRACT FROM AUTHOR]

We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function only through a single parameter (Euclidean space) or two parameters (spheres), and include the special boundary case of random Laplacian eigenfunctions.

Kac-Rice formulas express the expected number of elements a fiber of a random field has in terms of a multivariate integral. We consider here parametrized systems of polynomial equations that are linear in enough parameters, and provide a Kac-Rice formula for the expected number of solutions of the system when the parameters follow continuous distributions. Combined with Monte Carlo integration, we apply the formula to partition the parameter region according to the number of solutions or find a region in parameter space where the system has the maximal number of solutions. The motivation stems from the study of steady states of chemical reaction networks and gives new tools for the open problem of identifying the parameter region where the network has at least two positive steady states. We illustrate with numerous examples that our approach successfully handles a larger number of parameters than exact methods. [ABSTRACT FROM AUTHOR]

Non-convex optimization with local search heuristics has been widely used in machine learning, achieving many state-of-art results. It becomes increasingly important to understand why they can work for these NP-hard problems on typical data. The landscape of many objective functions in learning has been conjectured to have the geometric property that "all local optima are (approximately) global optima", and thus they can be solved efficiently by local search algorithms. However, establishing such property can be very difficult. In this paper, we analyze the optimization landscape of the random over-complete tensor decomposition problem, which has many applications in unsupervised learning, especially in learning latent variable models. In practice, it can be efficiently solved by gradient ascent on a non-convex objective. We show that for any small constant ε > 0 , among the set of points with function values (1 + ε) -factor larger than the expectation of the function, all the local maxima are approximate global maxima. Previously, the best-known result only characterizes the geometry in small neighborhoods around the true components. Our result implies that even with an initialization that is barely better than the random guess, the gradient ascent algorithm is guaranteed to solve this problem. However, achieving such a initialization with random guess would still require super-polynomial number of attempts. Our main technique uses Kac–Rice formula and random matrix theory. To our best knowledge, this is the first time when Kac–Rice formula is successfully applied to counting the number of local optima of a highly-structured random polynomial with dependent coefficients. [ABSTRACT FROM AUTHOR]

We prove a generalized Kac-Rice formula that, in a well defined regular setting, computes the expected cardinality of the preimage of a submanifold via a random map by expressing it as the integral of a density. Our proof starts from scratch and although it follows the guidelines of the standard proofs of Kac-Rice formula, it contains some new ideas coming from the point of view of measure theory. Generalizing further, we extend this formula to any other type of counting measure, such as the intersection degree. We discuss in depth the specialization to smooth Gaussian random sections of a vector bundle. Here, the formula computes the expected number of points where the section meets a given submanifold of the total space, it holds under natural non-degeneracy conditions and can be simplified by using appropriate connections. Moreover, we point out a class of submanifolds, that we call sub-Gaussian, for which the formula is locally finite and depends continuously with respect to the covariance of the first jet. In particular, this applies to any notion of singularity of sections that can be defined as the set of points where the jet prolongation meets a given semialgebraic submanifold of the jet space. Various examples of applications and special cases are discussed. In particular, we report a new proof of the Poincaré kinematic formula for homogeneous spaces and we observe how the formula simplifies for isotropic Gaussian fields on the sphere. [ABSTRACT FROM AUTHOR]

We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold h c for the magnitude of the external field necessary for trivialization and conjecture h c to be sharp. The Kac–Rice formula is our main tool. Our work extends Fyodorov [14], which identified the trivial regime for the special case of pure p-spin Hamiltonians with random external field. [ABSTRACT FROM AUTHOR]

This article introduces exact testing procedures on the mean of a Gaussian process X derived from the outcomes of ℓ 1 -minimization over the space of complex valued measures. The process X can be thought as the sum of two terms: first, the convolution between some kernel and a target atomic measure (mean of the process); second, a random perturbation by an additive centered Gaussian process. The first testing procedure considered is based on a dense sequence of grids on the index set of X and we establish that it converges (as the grid step tends to zero) to a randomized testing procedure: the decision of the test depends on the observation X and also on an independent random variable. The second testing procedure is based on the maxima and the Hessian of X in a grid-less manner. We show that both testing procedures can be performed when the variance is unknown (and the correlation function of X is known). These testing procedures can be used for the problem of deconvolution over the space of complex valued measures, and applications in frame of the Super-Resolution theory are presented. As a byproduct, numerical investigations may demonstrate that our grid-less method is more powerful (it detects sparse alternatives) than tests based on very thin grids. [ABSTRACT FROM AUTHOR]

We study the local repulsion between critical points of a stationary isotropic smooth planar Gaussian field. We show that the critical points can experience a soft repulsion which is maximal in the case of the random planar wave model, or a soft attraction of arbitrary high order. If the type of critical points is specified (extremum, saddle point), the points experience a hard local repulsion, that we quantify with the precise magnitude of the second factorial moment of the number of points in a small ball. [ABSTRACT FROM AUTHOR]

In this article, we consider the following family of random trigonometric polynomials p n (t , Y) = ∑ k = 1 n Y k 1 cos (k t) + Y k 2 sin (k t) for a given sequence of i.i.d. random variables Y k i , i ∈ { 1 , 2 } , k ≥ 1 , which are centered and standardized. We set N ([ 0 , π ] , Y) the number of real roots over [ 0 , π ] and N ([ 0 , π ] , G) the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin's condition on the distribution of the coefficients that lim n → ∞ Var N n ([ 0 , π ] , Y) n = lim n → ∞ Var N n ([ 0 , π ] , G) n + 1 30 E Y 1 1 4 - 3. The latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis. Actually, a more general result is proven in this article, which does not require that the coefficients are identically distributed. The proof mixes a recent result regarding Edgeworth expansions for distribution norms established in Bally et al. (Electron J Probab 23(45):1–51, 2018) with the celebrated Kac–Rice formula. [ABSTRACT FROM AUTHOR]

Let X be a complex projective manifold of dimension n defined over the reals, and let M denote its real locus. We study the vanishing locus Zsd in M of a random real holomorphic section sd of E ⊗Ld, where L → X is an ample line bundle and E → X is a rank r Hermitian bundle. When r ∈ {1, . . . ,n - 1}, we obtain an asymptotic of order dr-n2, as d goes to infinity, for the variance of the linear statistics associated with Zsd, including its volume. Given an open set U ⊂ M, we show that the probability that Zsd does not intersect U is a O of d -n2 when d goes to infinity. When n > 3, we also prove almost sure convergence for the linear statistics associated with a random sequence of sections of increasing degree. Our framework contains the case of random real algebraic submanifolds of RPn obtained as the common zero set of r independent Kostlan-Shub-Smale polynomials. [ABSTRACT FROM AUTHOR]

This paper aims at comparing theoretical approximations of the tail of the maximum of stochastic processes and the corresponding numerical evaluations. More particularly, we focus on the Pickands or double sum method, the Rice method, the Euler Characteristic method and a new one called the Poisson method. The numerical evaluation, performed using mainly Quasi Monte-Carlo integration and adaptations of the programs of Genz, show the domains of validity of each method. [ABSTRACT FROM AUTHOR]

We determine the asymptotic law for the fluctuations of the total number of critical points of random Gaussian spherical harmonics in the high degree limit. Our results have implications on the sophistication degree of an appropriate percolation process for modelling nodal domains of eigenfunctions on generic compact surfaces or billiards. [ABSTRACT FROM AUTHOR]

We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high-energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for the variances and we show that the empirical measures in the high-energy limits converge weakly to their expected values. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed. [ABSTRACT FROM AUTHOR]

Motivated by Wilmshurst's conjecture and more recent work of W. Li and A. Wei [17] , we determine asymptotics for the number of zeros of random harmonic polynomials sampled from the truncated model , recently proposed by J. Hauenstein, D. Mehta, and the authors [10] . Our results confirm (and sharpen) their ( 3 / 2 ) -powerlaw conjecture [10] that had been formulated on the basis of computer experiments; this outcome is in contrast with that of the model studied in [17] . For the truncated model we also observe a phase-transition in the complex plane for the Kac–Rice density. [ABSTRACT FROM AUTHOR]

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle $$E$$ over compact manifold $$M$$ canonically defines a metric on $$E$$ together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet theorem stating that if the bundle $$E$$ and the manifold $$M$$ are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble. [ABSTRACT FROM AUTHOR]

We establish the Central Limit Theorem, as the degree goes to infinity, for the normalized volume of the zero set of a rectangular Kostlan–Shub–Smale random polynomial system. This paper is a continuation of Central Limit Theorem for the number of real roots of Kostlan–Shub–Smale random polynomial systems by the same authors in which the case of square systems was considered. Our main tools are Kac-Rice formula and an expansion of the volume of the level set into the Itô-Wiener Chaos. [ABSTRACT FROM AUTHOR]

In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing. Now, existing models for the geometry and development of orientation preference maps in higher mammals make a crucial use of symmetry considerations. In this paper, we consider probabilistic models for V1 maps from the point of view of group theory; we focus on Gaussian random fields with symmetry properties and review the probabilistic arguments that allow one to estimate pinwheel densities and predict the observed value of p. Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry. We show that through the Plancherel decomposition of the space of complex-valued maps on the Euclidean plane, each infinite-dimensional irreducible unitary representation of the special Euclidean group yields a unique V1-like map, and we use representation theory as a symmetry-based toolbox to build orientation maps adapted to the most famous non- Euclidean geometries, viz. spherical and hyperbolic geometry. We find that most of the dominant traits of V1 maps are preserved in these; we also study the link between symmetry and the statistics of singularities in orientation maps, and show what the striking quantitative characteristics observed in animals become in our curved models. [ABSTRACT FROM AUTHOR]

In the primary visual cortex of many mammals, the processing of sensory information involves recognizing stimuli orientations. The repartition of preferred orientations of neurons in some areas is remarkable: a repetitive, non-periodic, layout. This repetitive pattern is understood to be fundamental for basic non-local aspects of vision, like the perception of contours, but important questions remain about its development and function. We focus here on Gaussian Random Fields, which provide a good description of the initial stage of orientation map development and, in spite of shortcomings we will recall, a computable framework for discussing general principles underlying the geometry of mature maps. We discuss the relationship between the notion of column spacing and the structure of correlation spectra; we prove formulas for the mean value and variance of column spacing, and we use numerical analysis of exact analytic formulae to study the variance. Referring to studies by Wolf, Geisel, Kaschube, Schnabel, and coworkers, we also show that spectral thinness is not an essential ingredient to obtain a pinwheel density of p, whereas it appears as a signature of Euclidean symmetry. The minimum variance property associated to thin spectra could be useful for information processing, provide optimal modularity for V1 hypercolumns, and be a first step toward a mathematical definition of hypercolumns. A measurement of this property in real maps is in principle possible, and comparison with the results in our paper could help establish the role of our minimum variance hypothesis in the development process. [ABSTRACT FROM AUTHOR]

In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a1sinhx+a2sinh2x+⋯+ansinhnx, where aj(j=1,2,…,n) are independent normally distributed random variables with mean zero and variance one, is (1/π)logn. This result is true for all K independent of x, provided K≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a1coshx+a2cosh2x+⋯+ancoshnx, with aj(j=1,2,…,n) as before, is also (1/π)logn.

This paper provides the asymptotic estimate for the expected number of real zeros of a random hyperbolic polynomial g1coshx+2g2cosh2x+…+ngncoshnx where gj,(j=1,2,…,n) are independent normally distributed random variables with mean zero and variance one. It is shown that for sufficiently large n this asymptotic value is (1/π)logn.

This article provides an asymptotic formula for the expected number of real zeros of a polynomial of the form [image omitted] for large n. The coefficients [image omitted] are assumed to be a sequence of dependent normally distributed random variables with E(aj(ω)) = 0, var(aj(ω)) = 1 and cov(ai(ω), aj(ω)) = ρ, 0 < ρ <1. We show that for the above dependent case this expected number is half that for the independent case. This behavior is similar to that of classical random algebraic polynomials. [ABSTRACT FROM AUTHOR]

We show that the expected number of real zeros of the nth degree polynomial with real independent identically distributed coefficients with common characteristic function Φ(z) = e-A(ln∣1/z∣)-a for 0< ∣z∣ < 1 and Φ(0) = 1, Φ(z) ≡0 for 1 ≦ ∣z∣ < ∞, with 1 < a and A ≧ a(a-1), is (a - 1)/(a - ½)log n asymptotically as n → ∞. [ABSTRACT FROM AUTHOR]

The expected number of real zeros of polynomials a0 + a1x + a2x2 +...+an-1xn-1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(ai, aj) = 1 - |i - j|/n, for i = 0,..., n - 1 and j = 0,..., n - 1, the above expected number of real zeros reduces significantly to O(log n)1/2. [ABSTRACT FROM AUTHOR]

This paper provides a formula to be used for obtaining the variance of the number of real zeros of random algebraic polynomial . The expected number of real zeros of this type of polynomial is known. An easy modification of this formula leads to a formula for the covariance for the number of real zeros in any two disjoint intervals. Using the latter, we show the covariance of the number of real zeros, in any two disjoint interval that can be obtained. To this end, we assume a normal standard distribution for the coefficients a j 's, j = 0, 1, 2,..., n . Although we give a formula for the variance, the evaluation of the asymptotic value for the variance remains our main task for future work. [ABSTRACT FROM AUTHOR]

This paper provides an asymptotic estimate for the expected number of level-crossings of random trigonometric polynomials a 0 (ω) + a 1 (ω)cosΘ + ··· + a n (ω)cos n Θ, where a j (ω), j = 0,..., n are k groups of independent normally distributed random variables with means μ i and variances , i = 1,..., k . It is shown that in order to obtain our result we need charac-teristics, namely μ i and , i = 1,..., k , to be any bounded absolute constants. [ABSTRACT FROM AUTHOR]

The asymptotic estimate for the expected number of real zeros of a random algebraic polynomial $$Q_n(x,\omega)=a_o(\omega){n\choose 0}+a_1(\omega){n\choose 1}x+a_2(\omega){n\choose 2}x^2+\cdots + a_n(\omega){n\choose n}x^n$$is known. The identical random coefficients a j(ω) are normally distributed defined on a probability space $$(\Omega, \Pr, \mathcal{A})$$, ω ∈Ω. The estimate for the expected number of zeros of the derivative of the above polynomial with respect to x is also known, which gives the expected number of maxima and minima of Q n( x, ω). In this paper we provide the asymptotic value for the expected number of zeros of the integration of Q n( x,ω) with respect to x. We give the geometric interpretation of our results and discuss the difficulties which arise when we consider a similar problem for the case of $$a_0(\omega)+a_1(\omega)x+a_2(\omega)x^2+\cdots + a_n(\omega)x^n$$. [ABSTRACT FROM AUTHOR]

Abstract: This paper provides a new result on the asymptotic estimate for the expected number of mml-level crossings of a random trigonometric polynomialmml l cos x + α2 cos 2x + . + αn cos nx, where mml, areindependent random variables with mean 0 and variance 1. It is shown that in the case of mml the actual error term is the same as in the case of K = 0 and is equal to O(1). [Copyright &y& Elsevier]

There is both mathematical and physical interest in the behaviour of the polynomial of the form $$a_0 + a_1 (_{\text{1}}^n {\kern 1pt} )^{1/2} x + a_2 (_{\text{2}}^n {\kern 1pt} )^{1/2} x^2 + \cdots + a_n (_n^n {\kern 1pt} )^{1/2} x^n $$ . The coefficients a j, j = 0,..., n are assumed to be independent normally distributed random variables with mean zero and variance σ2. In this paper by using the motion of exceedence measure for stochastic processes, for n large, we derive an asymptotic estimate for the expected area of the curve representing the above polynomial cut off by the x-axis. We show that our method can be used to obtain results for similar random polynomials. [ABSTRACT FROM AUTHOR]

This paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non-identical random coefficients. We assume that the coefficients {aj}n−1j=0 of the polynomial T(x)=a0+a1x+a2x2+⋯+an−1xn−1 are normally distributed, with mean E(aj)=μj+1, where μ≠0, and constant non-zero variance. It is shown that the behaviour of the random polynomial is independent of the variance on the interval (−1,1); it differs, however, for the cases of &z.sfnc;μ&z.sfnc;<1 and &z.sfnc;μ&z.sfnc;>1. On the intervals (−∞,−1) and (1,∞) we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance. Our result is consistent with those of previous works for identically distributed coefficients, in that the expected number of real zeros for μ≠0 is half of that for μ=0. [Copyright &y& Elsevier]