Learning multiple maps from conditional ordinal triplets

Singapore National Research Foundatio

Cognitive mechanisms of statistical learning and segmentation of continuous sensory input

Two classes of cognitive mechanisms have been proposed to explain segmentation of continuous sensory input into discrete recurrent constituents: clustering and boundary-finding mechanisms. Clustering mechanisms are based on identifying frequently co-occurring elements and merging them together as parts that form a single constituent. Bracketing (or boundary-finding) mechanisms work by identifying rarely co-occurring elements that correspond to the boundaries between discrete constituents. In a series of behavioral experiments, I tested which mechanisms are at play in the visual modality both during segmentation of a continuous syllabic sequence into discrete word-like constituents and during recognition of segmented constituents. Additionally, I explored conscious awareness of the products of statistical learning—whole constituents versus merged clusters of smaller subunits. My results suggest that both online segmentation and offline recognition of extracted constituents rely on detecting frequently co-occurring elements, a process likely based on associative memory. However, people are more aware of having learnt whole tokens than of recurrent composite clusters. © 2021, The Author(s)

Phylogenetic CSPs are Approximation Resistant

We study the approximability of a broad class of computational problems -- originally motivated in evolutionary biology and phylogenetic reconstruction -- concerning the aggregation of potentially inconsistent (local) information about $n$ items of interest, and we present optimal hardness of approximation results under the Unique Games Conjecture. The class of problems studied here can be described as Constraint Satisfaction Problems (CSPs) over infinite domains, where instead of values $\{0,1\}$ or a fixed-size domain, the variables can be mapped to any of the $n$ leaves of a phylogenetic tree. The topology of the tree then determines whether a given constraint on the variables is satisfied or not, and the resulting CSPs are called Phylogenetic CSPs. Prominent examples of Phylogenetic CSPs with a long history and applications in various disciplines include: Triplet Reconstruction, Quartet Reconstruction, Subtree Aggregation (Forbidden or Desired). For example, in Triplet Reconstruction, we are given $m$ triplets of the form $ij|k$ (indicating that ``items $i,j$ are more similar to each other than to $k$'') and we want to construct a hierarchical clustering on the $n$ items, that respects the constraints as much as possible. Despite more than four decades of research, the basic question of maximizing the number of satisfied constraints is not well-understood. The current best approximation is achieved by outputting a random tree (for triplets, this achieves a 1/3 approximation). Our main result is that every Phylogenetic CSP is approximation resistant, i.e., there is no polynomial-time algorithm that does asymptotically better than a (biased) random assignment. This is a generalization of the results in Guruswami, Hastad, Manokaran, Raghavendra, and Charikar (2011), who showed that ordering CSPs are approximation resistant (e.g., Max Acyclic Subgraph, Betweenness).Comment: 45 pages, 11 figures, Abstract shortened for arxi