arXiv subjects

Andreas Klingler

Publications and source records attributed to Andreas Klingler.

15 recordsLinked to original sources

arXiv2025

A New Approach to the Glass Transition of Percolated Polymers from the Perspective of Thermal Volume Expansion

In 1995, the Nobel Price winner P.W. Anderson made the following remarkable statement: The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and glass transition. Although there have been new theoretical developments in the meantime, in our opinion, this situation has only improved marginally to date. One of the main reasons is the insufficient consideration of experimental boundary conditions. A central experimental problem arises from the fact that the time constants required to achieve thermodynamic equilibrium increase sharply in the vicinity of and above a hypothetical static glass transition. If these equilibrium conditions are violated, additional internal thermodynamic variables come into play that normally alter the static and dynamic susceptibilities significantly and thus lead to misinterpretations of the experimental data. This raises, for example, the question of whether there are static property changes during the canonical glass transition and how these correlate with dynamic precursors. In this article, we attempt to find answers to these questions using a new experimental method called temperature-modulated optical refractometry in combination with the temperature-jump technique. The total time required to investigate the glass transition behaviour of our model epoxy was approximately two years.

cond-mat.soft↗

arXiv2025

From the dynamic to the static glass transition via hypersonic measurements using Brillouin spectroscopy

For the fragile, low-molecular-weight liquid diglycidyl ether of bisphenol A (DGEBA), we report on the dynamic glass transition and a further acoustic anomaly in the vicinity of the thermal glass transition based on hypersonic investigations of the longitudinal elastic modulus using Brillouin spectroscopy. This additional acoustic anomaly of the longitudinally polarized phonon is confirmed by the occurrence of an anomaly of the shear phonon at the same thermal glass transition temperature. Analysis of the generalized Cauchy relation suggests that both anomalies are coupled to a glass transition phenomenon independent of the so-called {\alpha}-relaxation process.

cond-mat.soft↗

arXiv2024

The Aldous$\unicode{x2013}$Hoover Theorem in Categorical Probability

The Aldous-Hoover Theorem concerns an infinite matrix of random variables whose distribution is invariant under finite permutations of rows and columns. It states that, up to equality in distribution, each random variable in the matrix can be expressed as a function only depending on four key variables: one common to the entire matrix, one that encodes information about its row, one that encodes information about its column, and a fourth one specific to the matrix entry. We state and prove the theorem within a category-theoretic approach to probability, namely the theory of Markov categories. This makes the proof more transparent and intuitive when compared to measure-theoretic ones. A key role is played by a newly identified categorical property, the Cauchy--Schwarz axiom, which also facilitates a new synthetic de Finetti Theorem. We further provide a variant of our proof using the ordered Markov property and the d-separation criterion, both generalized from Bayesian networks to Markov categories. We expect that this approach will facilitate a systematic development of more complex results in the future, such as categorical approaches to hierarchical exchangeability.

math.ST↗

arXiv2024

Polynomial decompositions with invariance and positivity inspired by tensors

We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we transfer results about decomposition structures, invariance under permutations of variables, positivity, rank inequalities and separations, approximations, and undecidability to real polynomials. Specifically, we define invariant decompositions of polynomials and characterize which polynomials admit such decompositions. We then include positivity: We define invariant separable and sum-of-squares decompositions, and characterize the polynomials similarly. We provide inequalities and separations between the ranks of the decompositions, and show that the separations are not robust with respect to approximations. For cyclically invariant decompositions, we show that it is undecidable whether the polynomial is nonnegative or sum-of-squares for all system sizes. Our work sheds new light on polynomials by putting them on an equal footing with tensors, and opens the door to extending this framework to other tensor product structures.

math-ph↗

arXiv2024

Positive Moments Forever: Undecidable and Decidable Cases

We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and undecidability for matrices over certain commutative and non-commutative polynomial rings. As consequences, we deduce that positivity is decidable for simple unitary linear recurrence sequences and undecidable for linear recurrence sequences over commutative polynomial rings. As a byproduct, we also prove a free version of Polya's theorem.

math.AG↗

arXiv2024

Homotopy Methods for Convex Optimization

Convex optimization encompasses a wide range of optimization problems that contain many efficiently solvable subclasses. Interior point methods are currently the state-of-the-art approach for solving such problems, particularly effective for classes like semidefinite programming, quadratic programming, and geometric programming. However, their success hinges on the construction of self-concordant barrier functions for feasible sets. In this work, we investigate and develop a homotopy-based approach to solve convex optimization problems. While homotopy methods have been considered in optimization before, their potential for general convex programs remains underexplored. This approach gradually transforms the feasible set of a trivial optimization problem into the target one while tracking solutions by solving a differential equation, in contrast to traditional central path methods. We establish a criterion that ensures that the homotopy method correctly solves the optimization problem and prove the existence of such homotopies for several important classes, including semidefinite and hyperbolic programs. Furthermore, we demonstrate that our approach numerically outperforms state-of-the-art methods in hyperbolic programming, highlighting its practical advantages.

math.OC↗

arXiv2024

Hidden Markov Models and the Bayes Filter in Categorical Probability

We use Markov categories to generalize the basic theory of Markov chains and hidden Markov models to an abstract setting. This comprises characterizations of hidden Markov models in terms of conditional independences and algorithms for Bayesian filtering and smoothing applicable in all Markov categories with conditionals. When instantiated in appropriate Markov categories, these algorithms specialize to existing ones such as the Kalman filter, forward-backward algorithm, and the Rauch-Tung-Striebel smoother. We also prove that the sequence of outputs of our abstract Bayes filter is itself a Markov chain with a concrete formula for its transition maps. There are two main features of this categorical framework. The first is its abstract generality, as manifested in our unified account of hidden Markov models and algorithms for filtering and smoothing in discrete probability, Gaussian probability, measure-theoretic probability, possibilistic nondeterminism and others at the same time. The second feature is the intuitive visual representation of information flow in terms of string diagrams.

math.ST↗

arXiv2023

Experimental determination of the glass transition temperature in a very narrow temperature range by Temperature Modulated Optical Refractometry

Latest since the landmark studies of Kovacs and co-workers on the glass transition of polymers, it is clear that thermally induced volume changes are of central importance for the understanding of the nature of the glass transition. Due to the kinetic background of the canonical (thermal) glass transition, it does not seem possible to derive a well-defined glass transition temperature T_g based on susceptibilities such as the thermal volume expansion coefficient, β(T), being strongly coupled to the glass transition process. Therefore, in practice, T_g is for example defined via the inflection point of the step-like β(T) curve. In this publication, we propose to use a thermo-optical feature, preceding the glass transition in the high-temperature phase, to determine the glass transition temperature T_g of a model polymer in a rather narrow temperature interval.

cond-mat.soft↗

arXiv2023

Approximate Completely Positive Semidefinite Factorizations and their Ranks

In this paper we show the existence of approximate completely positive semidefinite (cpsd) factorizations with a cpsd-rank bounded above (almost) independently from the cpsd-rank of the initial matrix. This is particularly relevant since the cpsd-rank of a matrix cannot, in general, be upper bounded by a function only depending on its size. For this purpose, we make use of the Approximate Caratheodory Theorem in order to construct an approximate matrix with a low-rank Gram representation. We then employ the Johnson-Lindenstrauss Lemma to improve to a logarithmic dependence of the cpsd-rank on the size.

math.AG↗

arXiv2023

Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations

The matrix rank and its positive versions are robust for small approximations, i.e. they do not decrease under small perturbations. In contrast, the multipartite tensor rank can collapse for arbitrarily small errors, i.e. there may be a gap between rank and border rank, leading to instabilities in the optimization over sets with fixed tensor rank. Can multipartite positive ranks also collapse for small perturbations? In this work, we prove that multipartite positive and invariant tensor decompositions exhibit gaps between rank and border rank, including tensor rank purifications and cyclic separable decompositions. We also prove a correspondence between positive decompositions and membership in certain sets of multipartite probability distributions, and leverage the gaps between rank and border rank to prove that these correlation sets are not closed. It follows that testing membership of probability distributions arising from resources like translational invariant Matrix Product States is impossible in finite time. Overall, this work sheds light on the instability of ranks and the unique behavior of bipartite systems.

quant-ph↗

arXiv2023

Many bounded versions of undecidable problems are NP-hard

Several physically inspired problems have been proven undecidable; examples are the spectral gap problem and the membership problem for quantum correlations. Most of these results rely on reductions from a handful of undecidable problems, such as the halting problem, the tiling problem, the Post correspondence problem or the matrix mortality problem. All these problems have a common property: they have an NP-hard bounded version. This work establishes a relation between undecidable unbounded problems and their bounded NP-hard versions. Specifically, we show that NP-hardness of a bounded version follows easily from the reduction of the unbounded problems. This leads to new and simpler proofs of the NP-hardness of bounded version of the Post correspondence problem, the matrix mortality problem, the positivity of matrix product operators, the reachability problem, the tiling problem, and the ground state energy problem. This work sheds light on the intractability of problems in theoretical physics and on the computational consequences of bounding a parameter.

quant-ph↗

arXiv2023

The d-separation criterion in Categorical Probability

The d-separation criterion detects the compatibility of a joint probability distribution with a directed acyclic graph through certain conditional independences. In this work, we study this problem in the context of categorical probability theory by introducing a categorical definition of causal models, a categorical notion of d-separation, and proving an abstract version of the d-separation criterion. This approach has two main benefits. First, categorical d-separation is a very intuitive criterion based on topological connectedness. Second, our results apply both to measure-theoretic probability (with standard Borel spaces) and beyond probability theory, including to deterministic and possibilistic networks. It therefore provides a clean proof of the equivalence of local and global Markov properties with causal compatibility for continuous and mixed random variables as well as deterministic and possibilistic variables.

math.ST↗

arXiv2022

Approximate Pythagoras Numbers on $*$-algebras over $\mathbb{C}$

The Pythagoras number of a sum of squares is the shortest length among its sums of squares representations. In many algebras, for example real polynomial algebras in two or more variables, there exists no upper bound on the Pythagoras number for all sums of squares. In this paper, we study how Pythagoras numbers in $*$-algebras over $\mathbb{C}$ behave with respect to small perturbations of elements. More precisely, the approximate Pythagoras number of an element is the smallest Pythagoras number among all elements in its $\varepsilon$-ball. We show that these approximate Pythagoras numbers are often significantly smaller than their exact versions, and allow for (almost) dimension-independent upper bounds. Our results use low-rank approximations for Gram matrices of sums of squares and estimates for the operator norm of the Gram map.

math.NT↗

arXiv2021

Approximate tensor decompositions: disappearance of many separations

It is well-known that tensor decompositions show separations, that is, that constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here we show that many of these separations disappear in the approximate case. Specifically, for every approximation error $\varepsilon$ and norm, we define the approximate rank as the minimum rank of an element in the $\varepsilon$-ball with respect to that norm. For positive semidefinite matrices, we show that the separations between rank, purification rank, and separable rank disappear for a large class of Schatten $p$-norms. For nonnegative tensors, we show that the separations between rank, positive semidefinite rank, and nonnegative rank disappear for all $\ell_p$-norms with $p>1$. For the trace norm ($p = 1$), we obtain upper bounds that depend on the ambient dimension. We also provide a deterministic algorithm to obtain the approximate decomposition attaining our bounds. Our main tool is an approximate version of Carathéodory's Theorem. Our results imply that many separations are not robust under small perturbations of the tensor, with implications in quantum many-body systems and communication complexity.

math.OC↗

arXiv2020

Cats climb entails mammals move: preserving hyponymy in compositional distributional semantics

To give vector-based representations of meaning more structure, one approach is to use positive semidefinite (psd) matrices. These allow us to model similarity of words as well as the hyponymy or is-a relationship. Psd matrices can be learnt relatively easily in a given vector space $M\otimes M^*$, but to compose words to form phrases and sentences, we need representations in larger spaces. In this paper, we introduce a generic way of composing the psd matrices corresponding to words. We propose that psd matrices for verbs, adjectives, and other functional words be lifted to completely positive (CP) maps that match their grammatical type. This lifting is carried out by our composition rule called Compression, Compr. In contrast to previous composition rules like Fuzz and Phaser (a.k.a. KMult and BMult), Compr preserves hyponymy. Mathematically, Compr is itself a CP map, and is therefore linear and generally non-commutative. We give a number of proposals for the structure of Compr, based on spiders, cups and caps, and generate a range of composition rules. We test these rules on a small sentence entailment dataset, and see some improvements over the performance of Fuzz and Phaser.

cs.CL↗