Identity types and weak factorization systems in Cauchy complete categories

It has been known that categorical interpretations of dependent type theory with Sigma- and Id-types induce weak factorization systems. When one has a weak factorization system (L, R) on a category C in hand, it is then natural to ask whether or not (L, R) harbors an interpretation of dependent type theory with Sigma- and Id- (and possibly Pi-) types. Using the framework of display map categories to phrase this question more precisely, one would ask whether or not there exists a class D of morphisms of C such that the retract closure of D is the class R and the pair (C, D) forms a display map category modeling Sigma- and Id- (and possibly Pi-) types. In this paper, we show, with the hypothesis that C is Cauchy complete, that there exists such a class D if and only if (C,R) itself forms a display map category modeling Sigma- and Id- (and possibly Pi-) types. Thus, we reduce the search space of our original question from a potentially proper class to a singleton.Comment: 14 page

Highly Robust Error Correction by Convex Programming

This paper discusses a stylized communications problem where one wishes to transmit a real-valued signal x ∈ ℝ^n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably when a fraction of the transmitted codeword is corrupted by arbitrary gross errors, and when in addition, all the entries of the codeword are contaminated by smaller errors (e.g., quantization errors). We show that if one encodes the information as Ax where A ∈ ℝ^(m x n) (m ≥ n) is a suitable coding matrix, there are two decoding schemes that allow the recovery of the block of n pieces of information x with nearly the same accuracy as if no gross errors occurred upon transmission (or equivalently as if one had an oracle supplying perfect information about the sites and amplitudes of the gross errors). Moreover, both decoding strategies are very concrete and only involve solving simple convex optimization programs, either a linear program or a second-order cone program. We complement our study with numerical simulations showing that the encoder/decoder pair performs remarkably well

Highly robust error correction by convex programming

This paper discusses a stylized communications problem where one wishes to transmit a real-valued signal x in R^n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably when a fraction of the transmitted codeword is corrupted by arbitrary gross errors, and when in addition, all the entries of the codeword are contaminated by smaller errors (e.g. quantization errors). We show that if one encodes the information as Ax where A is a suitable m by n coding matrix (m >= n), there are two decoding schemes that allow the recovery of the block of n pieces of information x with nearly the same accuracy as if no gross errors occur upon transmission (or equivalently as if one has an oracle supplying perfect information about the sites and amplitudes of the gross errors). Moreover, both decoding strategies are very concrete and only involve solving simple convex optimization programs, either a linear program or a second-order cone program. We complement our study with numerical simulations showing that the encoder/decoder pair performs remarkably well.Comment: 23 pages, 2 figure

Type-theoretic weak factorization systems

This article presents three characterizations of the weak factorization systems on finitely complete categories that interpret intensional dependent type theory with Sigma-, Pi-, and Id-types. The first characterization is that the weak factorization system (L,R) has the properties that L is stable under pullback along R and that all maps to a terminal object are in R. We call such weak factorization systems type-theoretic. The second is that the weak factorization system has an Id-presentation: roughly, it is generated by Id-types in the empty context. The third is that the weak factorization system (L, R) is generated by a Moore relation system, a generalization of the notion of Moore paths

Coinductive Control of Inductive Data Types

We combine the theory of inductive data types with the theory of universal measurings. By doing so, we find that many categories of algebras of endofunctors are actually enriched in the corresponding category of coalgebras of the same endofunctor. The enrichment captures all possible partial algebra homomorphisms, defined by measuring coalgebras. Thus this enriched category carries more information than the usual category of algebras which captures only total algebra homomorphisms. We specify new algebras besides the initial one using a generalization of the notion of initial algebra

Error correction and convex programming

This article discusses a recently proposed error correction method involving convex optimization [1]. From an encoded and corrupted real‐valued message, a receiver would like to determine the original message. A few entries of the encoded message are corrupted arbitrarily (which we call gross errors) and all the entries of the encoded message are corrupted slightly. We show that it is possible to recover the message with nearly the same accuracy as in the setting where no gross errors occur

Compressed Sensing with Coherent and Redundant Dictionaries

This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not only that compressed sensing is viable in this context, but also that accurate recovery is possible via an L1-analysis optimization problem. We introduce a condition on the measurement/sensing matrix, which is a natural generalization of the now well-known restricted isometry property, and which guarantees accurate recovery of signals that are nearly sparse in (possibly) highly overcomplete and coherent dictionaries. This condition imposes no incoherence restriction on the dictionary and our results may be the first of this kind. We discuss practical examples and the implications of our results on those applications, and complement our study by demonstrating the potential of L1-analysis for such problems

Coinductive control of inductive data types

We combine the theory of inductive data types with the theory of universal measurings. By doing so, we find that many categories of algebras of endofunctors are actually enriched in the corresponding category of coalgebras of the same endofunctor. The enrichment captures all possible partial algebra homomorphisms, defined by measuring coalgebras. Thus this enriched category carries more information than the usual category of algebras which captures only total algebra homomorphisms. We specify new algebras besides the initial one using a generalization of the notion of initial algebra.Comment: 21 page

Type theoretic weak factorization systems

This thesis presents a characterization of those categories with weak factorization systems that can interpret the theory of intensional dependent type theory with Σ, Π, and identity types. We use display map categories to serve as models of intensional dependent type theory. If a display map category (C, D) models Σ and identity types, then this structure generates a weak factorization system (L, R). Moreover, we show that if the underlying category C is Cauchy complete, then (C, R) is also a display map category modeling Σ and identity types (as well as Π types if (C, D) models Π types). Thus, our main result is to characterize display map categories (C, R) which model Σ and identity types and where R is part of a weak factorization system (L, R) on the category C. We offer three such characterizations and show that they are all equivalent when C has all finite limits. The first is that the weak factorization system (L, R) has the properties that L is stable under pullback along R and all maps to a terminal object are in R. We call such weak factorization systems type theoretic. The second is that the weak factorization system has what we call an Id-presentation: it can be built from certain categorical structure in the same way that a model of Σ and identity types generates a weak factorization system. The third is that the weak factorization system (L, R) is generated by a Moore relation system. This is a technical tool used to establish the equivalence between the first and second characterizations described. To conclude the thesis, we describe a certain class of convenient categories of topological spaces (a generalization of compactly generated weak Hausdorff spaces). We then construct a Moore relation system within these categories (and also within the topological topos) and thus show that these form display map categories with Σ and identity types (as well as Π types in the topological topos)