Sufficient Geometric Conditions Research Articles

Stationary Points of a Shallow Neural Network with Quadratic Activations and the Global Optimality of the Gradient Descent Algorithm

We consider the problem of training a shallow neural network with quadratic activation functions and the generalization power of such trained networks. Assuming that the samples are generated by a full rank matrix [Formula: see text] of the hidden network node weights, we obtain the following results. We establish that all full-rank approximately stationary solutions of the risk minimization problem are also approximate global optimums of the risk (in-sample and population). As a consequence, we establish that, when trained on polynomially many samples, the gradient descent algorithm converges to the global optimum of the risk minimization problem regardless of the width of the network when it is initialized at some value [Formula: see text], which we compute. Furthermore, the network produced by the gradient descent has a near zero generalization error. Next, we establish that initializing the gradient descent algorithm below [Formula: see text] is easily achieved when the weights of the ground truth matrix [Formula: see text] are randomly generated and the matrix is sufficiently overparameterized. Finally, we identify a simple necessary and sufficient geometric condition on the size of the training set under which any global minimizer of the empirical risk has necessarily zero generalization error. Funding: The research of E. C. Kizildag is supported by Columbia University, with the Distinguished Postdoctoral Fellowship in Statistics. Support from the National Science Foundation [Grant DMS-2015517] is gratefully acknowledged.

On a Sampling Problem for a Bargmann-Fock Space

The purpose of the present article is to provide geometric sufficient conditions for discrete points to be a sampling sequence for a generalized Hilbert Bargmann-Fock space in several complex variables.

Observability for Non-autonomous Systems

We study non-autonomous observation systems , where is a strongly measurable family of closed operators on a Banach space and is a family of bounded observation operators from to a Banach space . Based on an abstract uncertainty principle and a dissipation estimate, we prove that the observation system satisfies a final-state observability estimate in for measurable subsets . We present applications of the above result to families of uniformly strongly elliptic differential operators as well as non-autonomous Ornstein–Uhlenbeck operators on with observation operators . In the setting of non-autonomous strongly elliptic operators, we derive necessary and sufficient geometric conditions on the family of sets such that the corresponding observation system satisfies a final-state observability estimate.

A classification of combinatorial types of discriminantal arrangements

Manin and Schechtman introduced a family of arrangements of hyperplanes generalizing classical braid arrangements, which they called the $\textit{discriminantal arrangements}$. Athanasiadis proved a conjecture by Bayer and Brandt providing a full description of the combinatorics of discriminantal arrangements in the case of $\textit{very generic}$ arrangements. Libgober and Settepanella described a sufficient geometric condition for given arrangements to be $\textit{non very generic}$ in terms of the notion of dependency for a certain arrangement. Settepanella and the author generalized the notion of dependency introducing $r$-sets and $K_\mathbb{T}$-vector sets, and provided a sufficient condition for non very genericity but still not convenient to verify by hand. In this paper we give a classification of the $r$-sets, and a more explicit and tractable condition for non very genericity.

On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes

We study some geometric aspects of the higher order mean curvatures (or, more simply, the so-called $ r $-th mean curvatures) of a spacelike hypersurface immersed in a pp-wave spacetime, namely, in a connected Lorentzian manifold admitting a parallel and lightlike vector field. Initially, we develop general Minkowski-type integral formulas for compact (without boundary) spacelike hypersurfaces and we apply them to the study of the uniqueness and nonexistence of compact spacelike hypersurfaces in terms of their $ r $-mean curvatures. Next, we obtain a characterization of $ r $-stability for $ r $-maximal compact spacelike hypersurfaces through of the analysis of the first nonzero eigenvalue of an differential operator naturally attached to the $ r $-th mean curvature. For the noncompact case, by applying new forms of maximum principles on complete noncompact Riemannian manifolds due to Caminha [17] and Alías, Caminha and Nascimento [3], we obtain sufficient geometric conditions involving some $ r $-th mean curvature and the volume growth that allow us to establish some nonexistence results or to guarantee that a complete noncompact spacelike hypersurface is either totally geodesic, or totally umbilical, or maximal, or $ r $-maximal. We also obtain estimates for the index of minimum relative nullity of spacelike hypersurfaces.

A Geometric Sufficient Condition for Contact Wrench Feasibility

A fundamental problem in legged locomotion is to verify whether a desired trajectory satisfies all physical constraints, especially those for maintaining contacts. Although foot tipping can be avoided via the Zero Moment Point (ZMP) condition, preventing foot sliding and twisting leads to the more complex Contact Wrench Cone (CWC) constraints. This paper proposes an efficient algorithm to certify the inclusion of a net contact wrench in the CWC on flat ground with uniform friction. In addition to checking the ZMP criterion, the proposed method also verifies whether the linear force and the yaw moment are feasible. The key step in the algorithm is a novel exact geometric characterization of the yaw moment limits in the case when the support polygon is approximated by a single supporting line. We propose two approaches to select this approximating line, providing an accurate inner approximation of the ground truth yaw moment limits with only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ 18.80\%$</tex-math></inline-formula> (resp. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$7.13\%$</tex-math></inline-formula> ) error. The methods require only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1/150$</tex-math></inline-formula> (resp. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1/139$</tex-math></inline-formula> ) computation time compared to the exact CWC based on conic programming. As a benchmark, approximating the CWC using square friction pyramids requires similar computation time as the exact CWC, but has <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$&gt;\!19.35\%$</tex-math></inline-formula> error. Unlike the ZMP condition, our method provides a sufficient condition for contact wrench feasibility.

On planar polynomial geometric interpolation

In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the Höllig–Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.

SPECTRAL INEQUALITIES FOR COMBINATIONS OF HERMITE FUNCTIONS AND NULL-CONTROLLABILITY FOR EVOLUTION EQUATIONS ENJOYING GELFAND–SHILOV SMOOTHING EFFECTS

AbstractThis work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow us to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index $\frac {1}{2} \leq \mu &lt;1$ , we deduce sufficient geometric conditions on control subsets to ensure the null-controllability of evolution equations enjoying regularizing effects in the symmetric Gelfand–Shilov space $S^{\mu }_{\mu }(\mathbb {R}^{n})$ . These results apply in particular to derive the null-controllability in any positive time for evolution equations associated to certain classes of hypoelliptic non-self-adjoint quadratic operators, or to fractional harmonic oscillators.

Mathcal{N} = (1, 1) supersymmetric AdS3 in 10 dimensions

Warped AdS3 solutions in 10 dimensional supergravity that preserve mathcal{N} = (1, 1) supersymmetry are considered. Sufficient geometric conditions for their existence, and to stop the AdS3 factor experiencing an enhancement to AdS4, are presented. The internal space of such solutions decomposes as a foliation of M6 over an interval where M6 supports either an SU(3)- or SU(2)-structure. The former case is classified in terms of torsion classes and new solutions are found.

Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

We prove that the thickness property is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space ℝn. These equations are associated with operators of the form F(|Dx|), the function F : [0, + ∞) → ℝ being continuous and bounded from below. We also provide explicit feedbacks and constants associated with these stabilization properties. The notion of thickness is known to be a necessary and sufficient condition for the exact null-controllability of the fractional heat equations associated with the functions F(t) = t2s in the case s &gt; 1∕2. Our results apply in particular for this class of equations, but also for the half heat equation associated with the function F(t) = t, which is the most diffusive fractional heat equation for which exact null-controllability is known to fail from general thick control supports.

Risk-Perception-Aware Control Design Under Dynamic Spatial Risks

This letter proposes a novel Risk-Perception-Aware (RPA) control design using non-rational perception of risks associated with uncertain dynamic spatial costs. We use Cumulative Prospect Theory (CPT) to model the risk perception of a Decision Maker (DM) and use it to construct perceived risk functions that transform the uncertain dynamic spatial cost to deterministic perceived risks of a DM. These risks are then used to build safety sets which can represent risk-averse to risk-insensitive perception. Using these sets, we define novel notions of “inclusiveness” and “versatility” which can be employed to compare and evaluate any risk models in the context of RPA safety-critical controls. We then prove that CPT is the most “inclusive” and “versatile” model w.r.t. Conditional Value at Risk (CVaR) and Expected Risk (ER). Given a RPM, we construct a class of Control Barrier Functions (CBFs) and generate perceived-safety-critical controls using a Quadratic Program (QP) to guide an agent safely to a goal. For a class of truncated-Gaussian costs, we provide sufficient geometric conditions for the above QP to be feasible. We also prove that CPT-equipped RPA controller has both a larger feasible control set and more accurate stabilization w.r.t CVaR and ER models. We present simulations in a 2D environment to illustrate the proposed controller.

On the local in time well-posedness of an elliptic–parabolic ferroelectric phase-field model

We consider a state-of-the-art ferroelectric phase-field model arising from the engineering area in recent years, which is mathematically formulated as a coupled elliptic–parabolic differential system. We utilize the maximal parabolic regularity theory to show the local in time well-posedness of the ferroelectric problem in both 2D and 3D spaces, which is sharp in the sense that the local solution is unique and a blow-up criterion is present. The well-posedness result will firstly be proved under some general assumptions. Afterwards we give sufficient geometric and regularity conditions which will guarantee the fulfillment of the imposed assumptions.

Trace estimates of Toeplitz operators on Bergman spaces and applications to composition operators

Let $\Omega$ be a subdomain of $\mathbb{C}$ and let $\mu$ be a positive Borel measure on $\Omega$. In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operators $T\_\mu$ acting on Bergman spaces on $\Omega$. Let $(\lambda n(T\mu))$ be the decreasing sequence of the eigenvalues of $T\_\mu$, and let $\rho$ be an increasing function such that $\rho (n)/n^A$ is decreasing for some $A>0$. We give an explicit necessary and sufficient geometric condition on $\mu$ in order to have $\lambda n(T\mu)\asymp 1/\rho (n)$. As applications, we consider composition operators $C\_\varphi$, acting on some standard analytic spaces on the unit disc $\mathbb{D}$. First, we give a general criterion ensuring that the singular values of $C\_\varphi$ satisfy $s\_n(C\_\varphi ) \asymp 1/\rho(n)$. Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of $\varphi (\mathbb{D})$. We finally study the case where $\partial \varphi (\mathbb{D})$ meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of $h(T\_\mu)$, where $h$ is a suitable concave or convex function.

The geometry of generalized loxodromic elements

We explore geometric conditions which ensure that a given element of a finitely generated group is, or fails to be, generalized loxodromic; as part of this we prove a generalization of Sisto’s result that every generalized loxodromic element is Morse. We provide a sufficient geometric condition for an element of a small cancellation group to be generalized loxodromic in terms of the defining relations and provide a number of constructions which prove that this condition is sharp.

Enhance capability of separable ball criterion for the bipartite quantum states

The largest separable ball is a meaningful geometric sufficient condition for separability of bipartite quantum states which tells that all the states in the unit ball in Frobenius norm centered at the identity matrix are separable. In this paper, we propose an algorithm that can improve the capability of this criterion to detect the separability outside the unit ball by solving an optimization problem based on the invertible local operators, that is, to optimize with respect to the invertible square matrices A and B. The numerical examples have shown that our algorithm is more powerful than the original separable ball criterion.

Stabilization of wave equations on the torus with rough dampings

For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where $a(x) = \sum\_{j=1}^N a\_j 1\_{x\in R\_j}$ ($R\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\infty$ dampings. We show that this condition is always necessary for uniform stabilisation (for any compact (smooth) manifold and any $L^\infty$ damping), and we prove that it is sufficient in our particular case on $\mathbb{T}^2$ (and for our particular dampings).

Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein–Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein–Uhlenbeck operators, a necessary and sufficient condition for null-controllability on the moving control supports is established. The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are also given to ensure null-controllability.

Geometric conditions for the exact controllability of fractional free and harmonic Schrödinger equations

We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrodinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrodinger equations is derived from the Logvinenko–Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schrodinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel–Rotach formula.

Discrete Generalized Nonlinear Observer Canonical Form

In this paper, we define a discrete d-generalized nonlinear observer canonical form (GNOCF) that is more general than nonlinear observer canonical form (NOCF). Geometric necessary and sufficient conditions for the state equivalence to a d-GNOCF are found together with MATLAB programs. Our conditions with $$d=0$$ also give the geometric necessary and sufficient conditions for the state equivalence to a NOCF with output transformation (OT). Since our proofs are constructive, an OT and a state transformation can also be found in the theorem.

Geometric Control and Disturbance Decoupling for Fractional Systems

We develop a geometric approach for fractional linear time-invariant systems with Caputo-type derivatives. In particular, we generalize the fundamental notions of invariance and controlled invariance to the fractional setting. We then exploit this new geometric framework to address the disturbance decoupling problem via static pseudostate feedback, with and without stability. Our main contribution is a set of necessary and sufficient conditions for the disturbance decoupling problem that are related to the input-output properties of the closed-loop system, and hence they are applicable not just to Caputo-type derivatives but, more broadly, to any type of fractional system. These results show that, while the conditions for guaranteeing the existence of a decoupling pseudostate feedback remain essentially unchanged, the underlying theoretical framework is substantially different, because the fractional derivative is a nonlocal operator and this property plays a major role in the characterization of the evolution of the pseudostate trajectory. In particular, we show that, unlike the integer case, the infinite-dimensional nature of fractional systems means that feedback control is insufficient to maintain the pseudostate trajectory on a controlled invariant subspace, unless the entire past history of the pseudostate has evolved on that subspace. However, feedforward control can achieve this task under certain necessary and sufficient geometric conditions.