Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra

While computer programs and logical theories begin by declaring the concepts of interest, be it as data types or as predicates, network computation does not allow such global declarations, and requires *concept mining* and *concept analysis* to extract shared semantics for different network nodes. Powerful semantic analysis systems have been the drivers of nearly all paradigm shifts on the web. In categorical terms, most of them can be described as bicompletions of enriched matrices, generalizing the Dedekind-MacNeille-style completions from posets to suitably enriched categories. Yet it has been well known for more than 40 years that ordinary categories themselves in general do not permit such completions. Armed with this new semantical view of Dedekind-MacNeille completions, and of matrix bicompletions, we take another look at this ancient mystery. It turns out that simple categorical versions of the *limit superior* and *limit inferior* operations characterize a general notion of Dedekind-MacNeille completion, that seems to be appropriate for ordinary categories, and boils down to the more familiar enriched versions when the limits inferior and superior coincide. This explains away the apparent gap among the completions of ordinary categories, and broadens the path towards categorical concept mining and analysis, opened in previous work.Comment: 22 pages, 5 figures and 9 diagram

Generalized Low Rank Models

Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, $k$-means, $k$-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compressing, denoising, and imputing missing entries across all data types simultaneously. It also admits a number of interesting interpretations of the low rank factors, which allow clustering of examples or of features. We propose several parallel algorithms for fitting generalized low rank models, and describe implementations and numerical results.Comment: 84 pages, 19 figure

Quantum channels as a categorical completion

We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201

Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid Q, which we call `Q-order'. This requires a theory of semicategories enriched in the quantaloid Q, that admit a suitable Cauchy completion. There is a quantaloid Idl(Q) of Q-orders and ideal relations, and a locally ordered category Ord(Q) of Q-orders and monotone maps; actually, Ord(Q)=Map(Idl(Q)). In particular is Ord(Omega), with Omega a locale, the category of ordered objects in the topos of sheaves on Omega. In general Q-orders can equivalently be described as Cauchy complete categories enriched in the split-idempotent completion of Q. Applied to a locale Omega this generalizes and unifies previous treatments of (ordered) sheaves on Omega in terms of Omega-enriched structures.Comment: 21 page

Spectral C*-categories and Fell bundles with path-lifting

Following Crane's suggestion that categorification should be of fundamental importance in quantising gravity, we show that finite dimensional even $S^o$-real spectral triples over \bbc are already nothing more than full C*-categories together with a self-adjoint section of their domain and range maps, while the latter are equivalent to unital saturated Fell bundles over pair groupoids equipped with a path-lifting operator given by a normaliser. Interpretations can be made in the direction of quantum Higgs gravity. These geometries are automatically quantum geometries and we reconstruct the classical limit, that is, general relativity on a Riemannian spin manifold.Comment: 20 pages, 1 figur