We present a fast N log N time algorithm for computing quadratic-phase integrals. This three-parameter class of integrals models propagation in free space in the Fresnel approximation, passage through thin lenses, and propagation in quadratic graded-index media as well as any combination of any number of these and is therefore of importance in optics. By carefully managing the sampling rate, one need not choose N much larger than the space-bandwidth product of the signals, despite the highly oscillatory integral kernel. The only deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus the algorithm computes quadratic-phase integrals with a performance similar to that of the fast-Fourier-transform algorithm in computing the Fourier transform, in terms of both speed and accuracy. A quadratic-phase system is a unitary system whose output g͑u͒ is related to its input f͑u͒ through a quadratic-phase integral:where ␣, , and ␥ are real parameters. This relationship is also known by other names, including "linear canonical transforms." [1][2][3][4] This input-output relation can model a broad class of optical systems, including thin lenses, sections of free space in the Fresnel approximation, sections of quadratic graded-index media, and arbitrary concatenations of any number of these. 2,3Computation of the Fresnel diffraction integral, which is a special case of Eq. (1) with ␣ =  = ␥, has received the greatest attention because it describes the propagation of light in free space (see Refs. 5 and 6 and the references therein). As the input-output relationship represented by the Fresnel integral is space invariant and takes the form of a convolution, one can compute it by taking the Fourier transform (FT) of the input, multiplying by the transfer function of free space, and inverse transforming, with overall time ϳN log N. Here we present an algorithm with which to compute Eq. (1) in ϳN log N time, despite the fact that the relationship represented by the more general Eq. (1) is not space invariant and is not a convolution. It is important to underline that here N is chosen close to the space-bandwidth product of the set of input signals, which is usually the smallest possible value of N that can be chosen in terms of information-theoretic considerations. Therefore the algorithm is highly efficient and, although we do not present a formal proof of this, is quite possibly nearly the most efficient that is possible. Indeed, the distinguishing feature of the present approach is the care with which sampling and space-bandwidth product issues are handled. Straightforward use of conventional numerical methods can result in inefficiencies either because their complexity is larger than ϳN log N or because the highly oscillatory quadraticphase kernel in Eq. (1) forces N to be chosen much larger than the space-bandwidth product of the signals.In the method presented here, the samples of the output are computed in terms of the samples of the input, in the sa...
We report a fast and accurate algorithm for numerical computation of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs). Also known as quadratic-phase integrals, this class of integral transforms represents a broad class of optical systems including Fresnel propagation in free space, propagation in gradedindex media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic/astigmatic/non-orthogonal cases. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional case and the separable two-dimensional case. The algorithm takes ϳÑ log Ñ time, where Ñ is the two-dimensional space-bandwidth product of the signal. Our method properly tracks and controls the space-bandwidth products in two dimensions, in order to achieve information theoretically sufficient, but not wastefully redundant, sampling required for the reconstruction of the underlying continuous functions at any stage of the algorithm. Additionally, we provide an alternative definition of general 2D-NS-LCTs that shows its kernel explicitly in terms of its ten parameters, and relate these parameters bidirectionally to conventional ABCD matrix parameters.
Dense word embeddings, which encode meanings of words to low-dimensional vector spaces, have become very popular in natural language processing (NLP) research due to their state-of-the-art performances in many NLP tasks. Word embeddings are substantially successful in capturing semantic relations among words, so a meaningful semantic structure must be present in the respective vector spaces. However, in many cases, this semantic structure is broadly and heterogeneously distributed across the embedding dimensions making interpretation of dimensions a big challenge. In this study, we propose a statistical method to uncover the underlying latent semantic structure in the dense word embeddings. To perform our analysis, we introduce a new dataset (SEM-CAT) that contains more than 6500 words semantically grouped under 110 categories. We further propose a method to quantify the interpretability of the word embeddings. The proposed method is a practical alternative to the classical word intrusion test that requires human intervention.
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