On Clustering Images Using Compression

The need for the ability to cluster unknown data to better understand its relationship to know data is prevalent throughout science. Besides a better understanding of the data itself or learning about a new unknown object, cluster analysis can help with processing data, data standardization, and outlier detection. Most clustering algorithms are based on known features or expectations, such as the popular partition based, hierarchical, density-based, grid based, and model based algorithms. The choice of algorithm depends on many factors, including the type of data and the reason for clustering, nearly all rely on some known properties of the data being analyzed. Recently, Li et al. proposed a new universal similarity metric, this metric needs no prior knowledge about the object. Their similarity metric is based on the Kolmogorov Complexity of objects, the objects minimal description. While the Kolmogorov Complexity of an object is not computable, in "Clustering by Compression," Cilibrasi and Vitanyi use common compression algorithms to approximate the universal similarity metric and cluster objects with high success. Unfortunately, clustering using compression does not trivially extend to higher dimensions. Here we outline a method to adapt their procedure to images. We test these techniques on images of letters of the alphabet.

A Short History of Computational Complexity

National Science Foundation (CCR-998310); Army Research Office (DAAD19-02-1-0058)

On the Complexity of Quantum ACC

For any q > 1, let MOD_q be a quantum gate that determines if the number of 1's in the input is divisible by q. We show that for any q,t > 1, MOD_q is equivalent to MOD_t (up to constant depth). Based on the case q=2, Moore has shown that quantum analogs of AC^(0), ACC[q], and ACC, denoted QAC^(0)_wf, QACC[2], QACC respectively, define the same class of operators, leaving q > 2 as an open question. Our result resolves this question, implying that QAC^(0)_wf = QACC[q] = QACC for all q. We also prove the first upper bounds for QACC in terms of related language classes. We define classes of languages EQACC, NQACC (both for arbitrary complex amplitudes) and BQACC (for rational number amplitudes) and show that they are all contained in TC^(0). To do this, we show that a TC^(0) circuit can keep track of the amplitudes of the state resulting from the application of a QACC operator using a constant width polynomial size tensor sum. In order to accomplish this, we also show that TC^(0) can perform iterated addition and multiplication in certain field extensions.