Related papers: Representation Embeddings of Cartesian Theories

In this paper, we obtain a non-abelian analogue of Lubkin's embedding theorem for abelian categories. Our theorem faithfully embeds any small regular Mal'tsev category $\mathbb{C}$ in an $n$-th power of a particular locally finitely…

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented…

There is a large ongoing scientific effort in mechanistic interpretability to map embeddings and internal representations of AI systems into human-understandable concepts. A key element of this effort is the linear representation…

Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. We exhibit slices of the representation theory of $\Lambda$ that are always classifiable in stringent geometric terms. Namely, we prove that, for any…

Neural network representations are often analyzed as vectors in a fixed Euclidean space. However, their coordinates are not uniquely defined. If a hidden representation is transformed by an invertible linear map, the network function can be…

We describe the complete set of pairwise non-isomorphic irreducible modules S(a) over the algebra R given by the defining relation xy-yx=yy, and the rule how they could be glued to indecomposables. Namely, we show that Ext_k^1(S(a),S(b))=0,…

Partial combinatory algebras are algebraic structures that serve as generalized models of computation. In this paper, we study embeddings of pcas. In particular, we systematize the embeddings between relativizations of Kleene's models, of…

This paper presents a theory of systemic undecidability, reframing incomputability as a structural property of systems rather than a localized feature of specific functions or problems. We define a notion of causal embedding and prove a…

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame…

In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this…

We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify…

We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants…

When filtering a topological space by a single parameter, the theory of quiver representations provides a complete framework for decomposing the resulting persistence module to obtain its barcode. This is achieved by interpreting the…

We prove an extension of the theorem of Drinfeld, Grinberg and Kazhdan to arcs with arbitrary residue field. As an application we show that the embedding codimension is generically constant on each irreducible subset of the arc space which…

We introduce a nonsymmetric, associative tensor product among representations of Cuntz algebras by using embeddings. We show the decomposition formulae of tensor products for permutative representations explicitly We apply decomposition…

One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…

Relational representation learning transforms relational data into continuous and low-dimensional vector representations. However, vector-based representations fall short in capturing crucial properties of relational data that are complex…

The necessary and sufficient conditions for existence of a generalized representer theorem are presented for learning Hilbert space-valued functions. Representer theorems involving explicit basis functions and Reproducing Kernels are a…

This paper presents primarily two Euclidean embeddings of the quotient space generated by matrices that are identified modulo arbitrary row permutations. The original application is in deep learning on graphs where the learning task is…