arXiv subjects

Tobias Fritz

Publications and source records attributed to Tobias Fritz.

At least 37 records · Page 2Linked to original sources

arXiv2024

Intensive Care as One Big Sequence Modeling Problem

Reinforcement Learning in Healthcare is typically concerned with narrow self-contained tasks such as sepsis prediction or anesthesia control. However, previous research has demonstrated the potential of generalist models (the prime example being Large Language Models) to outperform task-specific approaches due to their capability for implicit transfer learning. To enable training of foundation models for Healthcare as well as leverage the capabilities of state of the art Transformer architectures, we propose the paradigm of Healthcare as Sequence Modeling, in which interaction between the patient and the healthcare provider is represented as an event stream and tasks like diagnosis and treatment selection are modeled as prediction of future events in the stream. To explore this paradigm experimentally we develop MIMIC-SEQ, a sequence modeling benchmark derived by translating heterogenous clinical records from MIMIC-IV dataset into a uniform event stream format, train a baseline model and explore its capabilities.

cs.LG↗

arXiv2024

Structural parameters and possible association of the Ultra-Faint Dwarfs Pegasus III and Pisces II from deep Hubble Space Telescope photometry

We present deep Hubble Space Telescope (HST) photometry of the ultra-faint dwarf (UFD) galaxies Pegasus III (Peg III) and Pisces II (Psc II), two of the most distant satellites in the halo of the Milky Way (MW). We measure the structure of both galaxies, derive mass-to-light ratios with newly determined absolute magnitudes, and compare our findings to expectations from UFD-mass simulations. For Peg III, we find an elliptical half-light radius of $a_h=1.88^{+0.42}_{-0.33}$ arcminutes ($118^{+31}_{-30}$ pc) and $M_V{=}{-4.17}^{+0.19}_{-0.22}$; for Psc II, we measure $a_h{=}1.31^{+0.10}_{-0.09}$ arcminutes ($69\pm8$ pc) and $M_V{=}{-4.28}^{+0.19}_{-0.16}$. We do not find any morphological features that indicate a significant interaction between the two has occurred, despite their close separation of only $\sim$40 kpc. Using proper motions (PMs) from Gaia early Data Release 3, we investigate the possibility of any past association by integrating orbits for the two UFDs in a MW-only and a combined MW and Large Magellanic Cloud (LMC) potential. We find that including the gravitational influence of the LMC is crucial, even for these outer-halo satellites, and that a possible orbital history exists where Peg III and Psc II experienced a close ($\sim$10-20 kpc) passage about each other just over $\sim$1 Gyr ago, followed by a collective passage around the LMC ($\sim$30-60 kpc) just under $\sim$1 Gyr ago. Considering the large uncertainties on the PMs and the restrictive priors imposed to derive them, improved PM measurements for Peg III and Psc II will be necessary to clarify their relationship. This would add to the rare findings of confirmed pairs of satellites within the Local Group.

astro-ph.GA↗

arXiv2024

Matrix majorization in large samples

One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Mu et al, we show that if certain monotones - namely multivariate extensions of Rényi divergences - are strictly ordered between the two tuples, then for sufficiently large $n$, there exists a stochastic matrix taking the $n$-fold Kronecker power of each input distribution to the $n$-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.

math.ST↗

arXiv2023

Characterizing the asymptotic and catalytic stochastic orders on topological abelian groups

We study the usual stochastic order between probability measures on preordered topological abelian groups, focusing on asymptotic and catalytic versions of the order. In the asymptotic version, a measure $μ$ dominates a measure $ν$ if the i.i.d.~random walk generated by $μ$ first-order dominates the one generated by $ν$ at late times. In the catalytic version, $μ$ dominates $ν$ if there is a third $τ$ such that the convolution $μ\ast τ$ first-order dominates $ν\ast τ$. Provided that the preorder on $G$ is induced by a suitably large positive cone and that both measures are compactly supported Radon, our main result gives a sufficient condition for asymptotic and catalytic dominance to hold in terms of a family of inequalities closely related to the cumulant-generating functions. While this sufficient condition requires these inequalities to be strict, the non-strict versions of these inequalities are easily seen to be necessary. In this sense, our result gives conditions that are necessary and sufficient in generic cases. This result has been known for $G = \mathbb{R}$, but is new already for $\mathbb{R}^n$ with $n > 1$. It is a direct application of a recently proven theorem of real algebra, namely a \emph{Vergleichsstellensatz} for preordered semirings. We finally use our result to derive a formula for the rate at which the probabilities of a random walk decay \emph{relative} to those of another, now for walks on a preordered topological vector space with compactly supported Radon steps. Taking one of these walks to be deterministic reproduces a version of Cramér's large deviation theorem for infinite dimensions.

math.PR↗

arXiv2023

Weak cartesian properties of simplicial sets

Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category $Δ$ to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in $Δ$ to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in $Δ$, which we characterize completely, along with several other classes of squares in $Δ$. Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example.

math.CT↗

arXiv2023

From Gs-monoidal to Oplax Cartesian Categories: Constructions and Functorial Completeness

Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal categories whose arrows are generalised relations, with enough structure to talk about domains and partial functions, but less structure than cartesian bicategories. The aim of this paper is threefold. The first goal is to extend the original definition of gs-monoidality by enriching it with a preorder on arrows, giving rise to what we call oplax cartesian categories. Second, we show that (preorder-enriched) gs-monoidal categories naturally arise both as Kleisli categories and as span categories, and the relation between the resulting formalisms is explored. Finally, we present two theorems concerning Yoneda embeddings on the one hand and functorial completeness on the other, the latter inducing a completeness result also for lax functors from oplax cartesian categories to $\mathbf{Rel}$.

cs.LO↗

arXiv2023

Weakly Markov categories and weakly affine monads

Introduced in the 1990s in the context of the algebraic approach to graph rewriting, gs-monoidal categories are symmetric monoidal categories where each object is equipped with the structure of a commutative comonoid. They arise for example as Kleisli categories of commutative monads on cartesian categories, and as such they provide a general framework for effectful computation. Recently proposed in the context of categorical probability, Markov categories are gs-monoidal categories where the monoidal unit is also terminal, and they arise for example as Kleisli categories of commutative affine monads, where affine means that the monad preserves the monoidal unit. The aim of this paper is to study a new condition on the gs-monoidal structure, resulting in the concept of weakly Markov categories, which is intermediate between gs-monoidal categories and Markov ones. In a weakly Markov category, the morphisms to the monoidal unit are not necessarily unique, but form a group. As we show, these categories exhibit a rich theory of conditional independence for morphisms, generalising the known theory for Markov categories. We also introduce the corresponding notion for commutative monads, which we call weakly affine, and for which we give two equivalent characterisations. The paper argues that these monads are relevant to the study of categorical probability. A case at hand is the monad of finite non-zero measures, which is weakly affine but not affine. Such structures allow to investigate probability without normalisation within an elegant categorical framework.

math.CT↗

arXiv2023

Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability

Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent conditions for second-order stochastic dominance, a widely used way of comparing probability distributions by their spread. Furthermore, we lay foundation for the theory of comparing statistical experiments within Markov categories by stating and proving the classical Blackwell-Sherman-Stein Theorem. Our version not only offers new insight into the proof, but its abstract nature also makes the result more general, automatically specializing to the standard Blackwell-Sherman-Stein Theorem in measure-theoretic probability as well as a Bayesian version that involves prior-dependent garbling. Along the way, we define and characterize representable Markov categories, within which one can talk about Markov kernels to or from spaces of distributions. We do so by exploring the relation between Markov categories and Kleisli categories of probability monads.

math.ST↗

arXiv2023

Partial Evaluations and the Compositional Structure of the Bar Construction

The algebraic expression $3 + 2 + 6$ can be evaluated to $11$, but it can also be partially evaluated to $5 + 6$. In categorical algebra, such partial evaluations can be defined in terms of the $1$-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on $\mathsf{Set}$ (which are those monads for which the underlying functor and the multiplication are weakly cartesian). We find that this is not true for all monads: we describe a finitary monad on $\mathsf{Set}$ for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all directed acyclic configurations in $Δ^n$, but generally not all inner horns.

math.CT↗

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↗

arXiv2023

Free gs-monoidal categories and free Markov categories

Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams ($=$term graphs) as a combinatorial data structure. We formulate the appropriate $2$-categorical universal property based on ideas of Walters and prove that our categories satisfy it. We expect our free categories to be relevant for computer implementations and we also argue that they can be used as statistical causal models generalizing Bayesian networks.

math.CT↗

arXiv2023

Abstract Vergleichsstellensätze for preordered semifields and semirings I

Real algebra is usually thought of as the study of certain kinds of preorders on fields and rings. Among its core themes are the separation theorems known as Positivstellensätze. However, there is a nascent subfield of real algebra which studies preordered semirings and semifields, which is motivated by applications to probability, graph theory and theoretical computer science, among others. Here, we contribute to this subfield by developing a number of foundational results for it, with two abstract Vergleichsstellensätze being our main theorems. Our first Vergleichsstellensatz states that every semifield preorder is the intersection of its total extensions. We apply this to derive our second main result, a Vergleichsstellensatz for certain non-Archimedean preordered semirings in which the homomorphisms to the tropical reals play an important role. We show how this result recovers the existing Vergleichsstellensatz of Strassen and (through the latter) the classical Positivstellensatz of Krivine--Kadison--Dubois.

math.RA↗

arXiv2022

Non-abelian and $\varepsilon$-curved homological algebra with arrow categories

Grandis's non-abelian homological algebra generalizes standard homological algebra in abelian categories to \textit{homological categories}, which are a broader class of categories including for example the category of lattices and Galois connections. Here, we prove that if $\mathsf{C}$ is any category with an ideal of null morphisms with respect to which (co)kernels exist, then the arrow category of $\mathsf{C}$ is a homological category. This broadens the applicability of Grandis's framework substantially. In particular, one can form the homology of chain complexes in $\mathsf{C}$ by taking the homology objects to be morphisms of $\mathsf{C}$, which one may think of as maps from an object of cycles to an object of chains modulo boundaries. One situation to which Grandis's original framework does not apply is \textit{$\varepsilon$-curved homological algebra}. This refers to chain complexes of normed spaces whose differential squares to zero only approximately, in the sense that $\|d^2\| \leq \varepsilon$ for some $\varepsilon > 0$. This is relevant for example in the theory of approximate representations of groups, where Kazhdan has successfully employed $\varepsilon$-curved homological techniques in an ad-hoc manner. We develop some basics of $\varepsilon$-curved homological algebra and note that our result on arrow categories facilitates the application of Grandis's theory.

math.KT↗

arXiv2022

Asymptotic and catalytic containment of representations of $\mathsf{SU}(n)$

Given two finite-dimensional representations $ρ$ and $σ$ of $\mathsf{SU}(n)$, when is there $n \in \mathbb{N}$ such that $ρ^{\otimes n}$ is isomorphic to a subrepresentation of $σ^{\otimes n}$? When is there a third representation $η$ such that $ρ\otimes η$ is a subrepresentation of $σ\otimes η$? We call these the questions of asymptotic and catalytic containment, respectively. We answer both questions in terms of an explicit family of inequalities. These inequalities are almost necessary and sufficient in the following sense. If two representations satisfy all inequalities strictly, then asymptotic and catalytic containment follow (the former in generic cases). Conversely, if asymptotic or catalytic containment holds, then the inequalities must hold non-strictly. These results are an instance of a recent \emph{Vergleichsstellensatz} applied to the representation semiring.

math.RT↗

arXiv2022

HubPUG: Proper Motions for Local Group Dwarfs observed with HST utilizing Gaia as a Reference Frame

We present the method behind HubPUG, a software tool built for recovering systemic proper motions (PMs) of Hubble Space Telescope (HST) fields with two epochs of observations by utilizing stars observed by Gaia as a foreground frame of reference. HST PM experiments have typically relied on the use of distant background galaxies or quasi-stellar objects (QSOs) as stationary sources against which to measure PMs. Without consistent profiles, background galaxies are more difficult to centroid, but benefit on-aggregate from their large numbers. QSOs, though they can be fit with stellar point-spread functions, are sparse, with most fields containing none. Historically, the use of stars as references against which to measure PMs would have been difficult because they have individual PMs of their own. However, Gaia has now provided positions and PMs for over 1.4 billion stars, which are much more likely to be well-imaged in the fields around targets versus background sources, have predictable stellar profiles, and require less observing time per-image for good signal-to-noise. This technique allows us to utilize the power of Gaia to measure the PM of targets too faint for Gaia to observe itself. We have recovered PMs for the Milky Way satellites Sculptor and Draco with comparable uncertainties over HST-only and Gaia-only measurements, limited primarily by the current capabilities of the Gaia data. We also show the promise of this method for satellites of M31 with a new PM measurement for Andromeda VII.

astro-ph.GA↗

arXiv2021

An Algebraic Approach to Physical Fields

According to the algebraic approach to spacetime, a thoroughgoing dynamicism, physical fields exist without an underlying manifold. This view is usually implemented by postulating an algebraic structure (e.g., commutative ring) of scalar-valued functions, which can be interpreted as representing a scalar field, and deriving other structures from it. In this work, we point out that this leads to the unjustified primacy of an undetermined scalar field. Instead, we propose to consider algebraic structures in which all (and only) physical fields are primitive. We explain how the theory of \emph{natural operations} in differential geometry -- the modern formalism behind classifying diffeomorphism-invariant constructions -- can be used to obtain concrete implementations of this idea for any given collection of fields. For concrete examples, we illustrate how our approach applies to a number of particular physical fields, including electrodynamics coupled to a Weyl spinor.

physics.hist-ph↗

arXiv2021

Star Formation Histories of Ultra-Faint Dwarf Galaxies: environmental differences between Magellanic and non-Magellanic satellites?

We present the color-magnitude diagrams and star formation histories (SFHs) of seven ultra-faint dwarf galaxies: Horologium 1, Hydra 2, Phoenix 2, Reticulum 2, Sagittarius 2, Triangulum 2, and Tucana 2, derived from high-precision Hubble Space Telescope photometry. We find that the SFH of each galaxy is consistent with them having created at least 80% of the stellar mass by $z\sim6$. For all galaxies, we find quenching times older than 11.5 Gyr ago, compatible with the scenario in which reionization suppresses the star formation of small dark matter halos. However, our analysis also reveals some differences in the SFHs of candidate Magellanic Cloud satellites, i.e., galaxies that are likely satellites of the Large Magellanic Cloud and that entered the Milky Way potential only recently. Indeed, Magellanic satellites show quenching times about 600 Myr more recent with respect to those of other Milky Way satellites, on average, even though the respective timings are still compatible within the errors. This finding is consistent with theoretical models that suggest that satellites' SFHs may depend on their host environment at early times, although we caution that within the error bars all galaxies in our sample are consistent with being quenched at a single epoch.

astro-ph.GA↗

arXiv2021

Probability, valuations, hyperspace: Three monads on Top and the support as a morphism

We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.

math.GN↗