In some examples, a method includes accessing input vectors in an input space, wherein the input vectors characterize elements of a physical system. The method may also include generating feature vectors from the input vectors, and the feature vectors are generated without any vector product operations between performed between any of the input vectors. An inner product of a pair of the feature vectors may correlate to an implicit kernel for the pair of feature vectors, and the implicit kernel may approximate a Gaussian kernel within a difference threshold. The method may further include providing the feature vectors to an application engine for use in analyzing the elements of the physical system, other elements in the physical system, or a combination of both.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A system comprising: an input engine to access characterizations of elements of a physical system, the characterizations as input vectors in an input space; a mapping engine to generate feature vectors in a feature space from the input vectors, wherein an inner product of a pair of the feature vectors correlates to an implicit kernel for the pair of feature vectors and the implicit kernel approximates a Gaussian kernel within a difference threshold, and wherein generation of the feature vectors comprises: determination of a concomitant rank order (CRO) hash set of a particular input vector used to generate to a corresponding feature vector; assignment of a non-zero value to vector elements of the corresponding feature vector at vector indices represented by hash values of the CRO hash set; and an application engine to utilize the feature vectors generated from the input vectors to operate on the elements of the physical system, other elements of the physical system, or a combination of both. 2. The system of claim 1, wherein the mapping engine is further to generate the feature vectors through: assignment of a zero value to vector elements of the corresponding feature vector at vector indices not represented by the hash values of the CRO hash set. 3. The system of claim 1, wherein the mapping engine is to assign the non-zero value as a ‘1’ value to vector elements of the feature vector; and wherein the feature vectors generated by the mapping engine are binary vectors. 4. The system of claim 1, wherein the feature vectors generated by the mapping engine are sparse vectors with a ratio of non-zero vector elements to total vector elements that is less than a sparsity threshold. 5. The system of claim 1, wherein the feature vectors generated by the mapping engine are high-dimensional vectors with a total number of vector elements that exceeds a high-dimension threshold. 6. The system of claim 1, wherein the application engine comprises a linear classifier, a clustering engine, a regression engine, or any combination thereof. 7. A method comprising: accessing input vectors in an input space, the input vectors characterizing elements of a physical system; generating feature vectors from the input vectors, wherein: an inner product of a pair of the feature vectors correlates to an implicit kernel for the pair of feature vectors; the implicit kernel approximates a Gaussian kernel within a difference threshold; and the feature vectors are generated without any vector product operations performed between any of the input vectors; and providing the feature vectors to an application engine for use in analyzing the elements of the physical system, other elements in the physical system, or a combination of both. 8. The method of claim 7, wherein the generating comprises: accessing a dimensionality parameter and a hash numeral parameter; for each input vector of the input vectors: determining a concomitant rank order (CRO) hash set for the input vector with a number of hash values equal to the hash numeral parameter; generating a corresponding feature vector for the input vector with a vector size equal to the dimensional parameter; and assigning a ‘1’ value for vector elements of the corresponding feature vector with vector indices equal to the hash values of the CRO hash set and assigning a ‘0’ value for other vector elements of the feature vector. 9. The method of claim 8, wherein the dimensionality parameter exceeds a high-dimension threshold. 10. The method of claim 8, wherein a ratio between the hash numeral parameter and the dimensionality parameter is less than a sparsity threshold; and wherein the corresponding feature vectors are sparse binary feature vectors. 11. The method of claim 7, wherein the application engine comprises a linear classifier; and wherein providing comprises providing the feature vectors to the linear classifier to train an application model for classifying the elements of the physical system. 12. The method of claim 7, wherein the application engine comprises a clustering engine; and wherein providing comprises providing the feature vectors to the clustering engine to cluster the elements of the physical system. 13. The method of claim 7, wherein the application engine comprises a regression engine; and wherein providing comprises providing the feature vectors to the regression engine to perform a regression analysis for the elements of the physical system. 14. A non-transitory machine-readable medium comprising instructions executable by a processing resource to: access input vectors in an input space, the input vectors characterizing elements of a physical system; generate, from the input vectors, sparse binary feature vectors in a feature space, wherein: an inner product of a pair of the generated sparse binary feature vectors correlates to an implicit kernel for the pair and the implicit kernel approximates a Gaussian kernel within a difference threshold; generation of each sparse binary feature vector is performed without any vector product operations, and comprises: determination of a concomitant rank order (CRO) hash set for an input vector corresponding to the sparse binary feature vector; assignment of a ‘1’ value for vector elements of the sparse binary feature vector with vector indices equal to hash values of the CRO hash set; and assignment of a ‘0’ value for other vector elements of the sparse binary feature vector; and provide the sparse binary feature vectors to an application engine for use in analyzing the elements of the physical system, other elements of the physical system, or a combination of both. 15. The non-transitory machine-readable medium of claim 14, wherein each of the sparse binary feature vectors is sparse by having a ratio of vector elements with a ‘1’ value to total vector elements that is less than a sparsity threshold.