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Real-world data typically contain repeated and periodic patterns. Thissuggests that they can be effectively represented and compressed using only afew coefficients of an appropriate basis (e.g., Fourier, Wavelets, etc.).However, distance estimation when the data are represented using different setsof coefficients is still a largely unexplored area. This work studies theoptimization problems related to obtaining the \emph{tightest} lower/upperbound on Euclidean distances when each data object is potentially compressedusing a different set of orthonormal coefficients. Our technique leads totighter distance estimates, which translates into more accurate search,learning and mining operations \textit{directly} in the compressed domain.  We formulate the problem of estimating lower/upper distance bounds as anoptimization problem. We establish the properties of optimal solutions, andleverage the theoretical analysis to develop a fast algorithm to obtain an\emph{exact} solution to the problem. The suggested solution provides thetightest estimation of the $L_2$-norm or the correlation. We show that typicaldata-analysis operations, such as k-NN search or k-Means clustering, canoperate more accurately using the proposed compression and distancereconstruction technique. We compare it with many other prevalent compressionand reconstruction techniques, including random projections and PCA-basedtechniques. We highlight a surprising result, namely that when the data arehighly sparse in some basis, our technique may even outperform PCA-basedcompression.  The contributions of this work are generic as our methodology is applicableto any sequential or high-dimensional data as well as to any orthogonal datatransformation used for the underlying data compression scheme.

Compressive Mining: Fast and Optimal Data Mining in the Compressed Domain