Modifying Hamming Spaces for Efficient Search

We focus on the efficient search for the most similar bit strings to a given query in the Hamming space. The distance of this space can be lower-bounded by a function based on a difference of the number of ones in the compared strings, i.e. their weights. Recently, such property has been successfully used by the Hamming Weight Tree (HWT) indexing structure. We propose modifications of the bit strings that preserve pairwise Hamming distances but improve the tightness of these lower bounds, so the query evaluation with the HWT is several times faster. We also show that the unbalanced bit strings, recently reported to provide similar quality of search as the traditionally used balanced bit strings, are more easy to index with the HWT. Combined with the distance preserving modifications, the HWT query evaluation can be more than one order of magnitude faster than the HWT baseline. Finally, we show that such modifications are useful even for a very complex data where the search with the HWT is slower than a sequential search

Adding Cues to Binary Feature Descriptors for Visual Place Recognition

In this paper we propose an approach to embed continuous and selector cues in binary feature descriptors used for visual place recognition. The embedding is achieved by extending each feature descriptor with a binary string that encodes a cue and supports the Hamming distance metric. Augmenting the descriptors in such a way has the advantage of being transparent to the procedure used to compare them. We present two concrete applications of our methodology, demonstrating the two considered types of cues. In addition to that, we conducted on these applications a broad quantitative and comparative evaluation covering five benchmark datasets and several state-of-the-art image retrieval approaches in combination with various binary descriptor types.Comment: 8 pages, 8 figures, source: www.gitlab.com/srrg-software/srrg_bench, submitted to ICRA 201

Static Data Structure Lower Bounds Imply Rigidity

We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of $t \geq \omega(\log^2 n)$ on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space $(s= (1+\varepsilon)n)$, would already imply a semi-explicit ($\bf P^{NP}\rm$) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial ($t\geq n^{\delta}$) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime $(s=n+o(n))$, we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest

Quantum Error Correction via Codes over GF(4)

The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information Theory. Replaced Sept. 24, 1996, to correct a number of minor errors. Replaced Sept. 10, 1997. The second section has been completely rewritten, and should hopefully be much clearer. We have also added a new section discussing the developments of the past year. Finally, we again corrected a number of minor error