Deep Unsupervised Similarity Learning using Partially Ordered Sets

Unsupervised learning of visual similarities is of paramount importance to computer vision, particularly due to lacking training data for fine-grained similarities. Deep learning of similarities is often based on relationships between pairs or triplets of samples. Many of these relations are unreliable and mutually contradicting, implying inconsistencies when trained without supervision information that relates different tuples or triplets to each other. To overcome this problem, we use local estimates of reliable (dis-)similarities to initially group samples into compact surrogate classes and use local partial orders of samples to classes to link classes to each other. Similarity learning is then formulated as a partial ordering task with soft correspondences of all samples to classes. Adopting a strategy of self-supervision, a CNN is trained to optimally represent samples in a mutually consistent manner while updating the classes. The similarity learning and grouping procedure are integrated in a single model and optimized jointly. The proposed unsupervised approach shows competitive performance on detailed pose estimation and object classification.Comment: Accepted for publication at IEEE Computer Vision and Pattern Recognition 201

Complexity of matrix problems

In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are called wild problems. We show in an explicit form that this problem contains all classification matrix problems given by quivers or posets. Then we prove that it does not contain (but is contained in) the problem of classifying three-valent tensors. Hence, all wild classification problems given by quivers or posets have the same complexity; moreover, a solution of any one of these problems implies a solution of each of the others. The problem of classifying three-valent tensors is more complicated.Comment: 24 page

Quasiplanar diagrams and slim semimodular lattices

A (Hasse) diagram of a finite partially ordered set (poset) P will be called quasiplanar if for any two incomparable elements u and v, either v is on the left of all maximal chains containing u, or v is on the right of all these chains. Every planar diagram is quasiplanar, and P has a quasiplanar diagram iff its order dimension is at most 2. A finite lattice is slim if it is join-generated by the union of two chains. We are interested in diagrams only up to similarity. The main result gives a bijection between the set of the (similarity classes of) finite quasiplanar diagrams and that of the (similarity classes of) planar diagrams of finite, slim, semimodular lattices. This bijection allows one to describe finite posets of order dimension at most 2 by finite, slim, semimodular lattices, and conversely. As a corollary, we obtain that there are exactly (n-2)! quasiplanar diagrams of size n.Comment: 19 pages, 3 figure

Expressive Logics for Coinductive Predicates

The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata

Degeneration and orbits of tuples and subgroups in an Abelian group

A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained.Comment: 13 pages, 1 figur