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Haim Horowitz

Publications and source records attributed to Haim Horowitz.

At least 19 recordsLinked to original sources
arXiv2023

Abstract Corrected Iterations

We consider $(<\lambda)$-support iterations of a version of $(<\lambda)$-strategically complete $\lambda^+$-c.c. definable forcing notions along partial orders. We show that such iterations can be corrected to yield an analog of a result by Judah and Shelah for finite support iterations of Suslin ccc forcing, namely that if $(\mathbb{P}_{\alpha}, \underset{\sim}{\mathbb Q_{\beta}} : \alpha \leq \delta, \beta <\delta)$ is a FS iteration of Suslin ccc forcing and $U\subseteq \delta$ is sufficiently closed, then letting $\mathbb{P}_U$ be the iteration along $U$, we have $\mathbb{P}_U \lessdot \mathbb{P}_{\delta}$.

math.LO↗
arXiv2022

Quantum Variational Rewinding for Time Series Anomaly Detection

Electron dynamics, financial markets and nuclear fission reactors, though seemingly unrelated, all produce observable characteristics evolving with time. Within this broad scope, departures from normal temporal behavior range from academically interesting to potentially catastrophic. New algorithms for time series anomaly detection (TAD) are therefore certainly in demand. With the advent of newly accessible quantum processing units (QPUs), exploring a quantum approach to TAD is now relevant and is the topic of this work. Our approach - Quantum Variational Rewinding, or, QVR - trains a family of parameterized unitary time-devolution operators to cluster normal time series instances encoded within quantum states. Unseen time series are assigned an anomaly score based upon their distance from the cluster center, which, beyond a given threshold, classifies anomalous behavior. After a first demonstration with a simple and didactic case, QVR is used to study the real problem of identifying anomalous behavior in cryptocurrency market data. Finally, multivariate time series from the cryptocurrency use case are studied using IBM's Falcon r5.11H family of superconducting transmon QPUs, where anomaly score errors resulting from hardware noise are shown to be reducible by as much as 20% using advanced error mitigation techniques.

quant-ph↗
arXiv2022

A quantum generative model for multi-dimensional time series using Hamiltonian learning

Synthetic data generation has proven to be a promising solution for addressing data availability issues in various domains. Even more challenging is the generation of synthetic time series data, where one has to preserve temporal dynamics, i.e., the generated time series must respect the original relationships between variables across time. Recently proposed techniques such as generative adversarial networks (GANs) and quantum-GANs lack the ability to attend to the time series specific temporal correlations adequately. We propose using the inherent nature of quantum computers to simulate quantum dynamics as a technique to encode such features. We start by assuming that a given time series can be generated by a quantum process, after which we proceed to learn that quantum process using quantum machine learning. We then use the learned model to generate out-of-sample time series and show that it captures unique and complex features of the learned time series. We also study the class of time series that can be modeled using this technique. Finally, we experimentally demonstrate the proposed algorithm on an 11-qubit trapped-ion quantum machine.

quant-ph↗
arXiv2019

Turing Invariant Sets and the Perfect Set Property

We show that $ZF+DC+$"all Turing invariant sets of reals have the perfect set property" implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.

math.LO↗
arXiv2019

Compact Sets of Baire Class One Functions and Maximal Almost Disjoint Families

We provide a proof that analytic almost disjoint families of infinite sets of integers cannot be maximal using a result of Bourgain about compact sets of Baire class one functions. Inspired by this and related ideas, we then provide a new proof of that there are no maximal almost disjoint families in Solovay's model. We then use the ideas behind this proof to provide an extension of a dichotomy result by Rosenthal and by Bourgain, Fremlin and Talagrand to general pointwise bounded functions in Solovay's model. We then show that the same conclusions can be drawn about the model obtained when we add a generic selective ultrafilter to the Solovay model.

math.LO↗
arXiv2018

On the definability of mad families of vector spaces

We consider the definability of mad families in vector spaces of the form $\underset{n<\omega}{\bigoplus} F$ where $F$ is a field of cardinality $\leq \aleph_0$. We show that there is no analytic mad family of subspaces when $F=\mathbb{F}_2$, partially answering a question of Smythe. Our proof relies on a variant of Mathias forcing restricted to a certain idempotent ultrafilter whose existence follows from Glazer's proof of Hindman's theorem.

math.LO↗
arXiv2018

$\kappa$-Madness and Definability

Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal $\kappa$, there are no $\Sigma^1_1(\kappa)-\kappa-$mad families.

math.LO↗
arXiv2017

Madness and regularity properties

Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $\omega^{\omega}$-bounding sufficiently absolute forcing notions $\mathbb Q$.

math.LO↗
arXiv2017

Mad families and non-meager filters

We prove the consistency of ZF+DC+"there are no mad families"+"there exists a non-meager filter on $\omega$" relative to ZFC, answering a question of Neeman and Norwood. We also introduce a weaker version of madness, and we strengthen the result from [HwSh:1090] by showing that no such families exist in our model.

math.LO↗
arXiv2016

On the classification of definable ccc forcing notions

We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, ``$\text{ZF}+\text{DC}_{\omega_1}+$all sets of reals are $I_{\mathbb Q,\aleph_0}$-measurable'' implies the existence of an inner model with a measurable cardinal. We also introduce a wide class of Suslin ccc forcing notions which add a Hechler real, so that the above result applies to them.

math.LO↗
arXiv2016

Saccharinity with ccc

Using creature technology, we construct families of Suslin ccc non-sweet forcing notions $\mathbb Q$ such that $ZFC$ is equiconsistent with $ZF+$"every set of reals equals a Borel set modulo the $(\leq \aleph_1)$-closure of the null ideal associated with $\mathbb Q$"+"there is an $\omega_1$-sequence of distinct reals". This answers a question of the second author and Kellner. As an application of independent interest, we also show how our forcing adds a new $\Pi^1_2$ singleton over $L$ without relying on $L$-combinatorics.

math.LO↗