Categorical invariance and structural complexity in human concept learning

An alternative account of human concept learning based on an invariance measure of the categorical\ud stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural\ud complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or\ud size. To do this we introduce a mathematical framework based on the notion of a Boolean differential\ud operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the\ud category in respect to its dimensions. Using this framework, we propose that the structural complexity\ud of a Boolean category is indirectly proportional to its degree of categorical invariance and directly\ud proportional to its cardinality or size. Consequently, complexity and invariance notions are formally\ud unified to account for concept learning difficulty. Beyond developing the above unifying mathematical\ud framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of\ud the SHJ [Shepard, R. N., Hovland, C. L.,&Jenkins, H. M. (1961). Learning and memorization of classifications.\ud Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three\ud binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the\ud degree of learning difficulty of a large class of categories (in particular, the 41 category types studied\ud by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature,\ud 407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of\ud categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g.,\ud cognitively tractable)

Maximal equicontinuous factors and cohomology for tiling spaces

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that this map is injective in degree one and has torsion free cokernel. We show by example, however, that the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology

Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays

Understanding the local behaviour of structured multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms, and specifically property testers, to understand local properties of the data. We focus on the natural local problem of testing pattern freeness: given a large $d$-dimensional array $A$ and a fixed $d$-dimensional pattern $P$ over a finite alphabet, we say that $A$ is $P$-free if it does not contain a copy of the forbidden pattern $P$ as a consecutive subarray. The distance of $A$ to $P$-freeness is the fraction of entries of $A$ that need to be modified to make it $P$-free. For any $\epsilon \in [0,1]$ and any large enough pattern $P$ over any alphabet, other than a very small set of exceptional patterns, we design a tolerant tester that distinguishes between the case that the distance is at least $\epsilon$ and the case that it is at most $a_d \epsilon$, with query complexity and running time $c_d \epsilon^{-1}$, where $a_d < 1$ and $c_d$ depend only on $d$. To analyze the testers we establish several combinatorial results, including the following $d$-dimensional modification lemma, which might be of independent interest: for any large enough pattern $P$ over any alphabet (excluding a small set of exceptional patterns for the binary case), and any array $A$ containing a copy of $P$, one can delete this copy by modifying one of its locations without creating new $P$-copies in $A$. Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data. They serve as a first step towards a general understanding of local properties of multi-dimensional arrays, as any such property can be characterized by a fixed family of forbidden patterns

Some results on 2n−m2^{n-m} designs of resolution IV with (weak) minimum aberration

It is known that all resolution IV regular $2^{n-m}$ designs of run size $N=2^{n-m}$ where $5N/16<n<N/2$ must be projections of the maximal even design with $N/2$ factors and, therefore, are even designs. This paper derives a general and explicit relationship between the wordlength pattern of any even $2^{n-m}$ design and that of its complement in the maximal even design. Using these identities, we identify some (weak) minimum aberration $2^{n-m}$ designs of resolution IV and the structures of their complementary designs. Based on these results, several families of minimum aberration $2^{n-m}$ designs of resolution IV are constructed.Comment: Published in at http://dx.doi.org/10.1214/08-AOS670 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org