{'fraction_non_alphanumeric': 0.012154696132596685, 'fraction_numerical': 0.0011049723756906078, 'mean_word_length': 5.163265306122449, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Multiple local optima in vector quantizers

Two results are presented on vector quantizers meeting necessary conditions for optimality. First a simple generalization of well-known centroid and moment properties of the squared-error distortion measure to a weighted quadratic distortion measure with an input dependent weighting is presented. The second result is an application of the squared-error special case of the first result to a simulation study of the design of 1 bit per sample two- and three-dimensional quantizers for a memoryless Gaussian source using the generalized Lloyd technique. The existence of multiple distinct local optima is demonstrated, thereby showing that sufficient conditions for unique local optima do not exist for this simple common case. It is also shown that at least three dimensions are required for a vector quantizer to outperform a scalar quantizer for this source.