Lower-dimensional subspace approximation of a dataset

1. A computer-implemented method for performing semantic document analysis, the method comprising:

determining a semantic analysis matrix corresponding to a set of documents, wherein each entry of the semantic analysis matrix indicates a number of times a respective term of a set of terms appears in a corresponding document in the set of documents;

determining a lower-dimensional compressed matrix from the semantic analysis matrix, wherein determining the lower-dimensional compressed matrix comprises;

determining a set of leverage scores corresponding to rows of the semantic analysis matrix; and

sampling a first subset of the rows of the semantic analysis matrix based at least in part on a corresponding subset of the set of leverage scores;

determining a target rank parameter;

determining an accuracy tolerance;

determining, based at least in part on the target rank parameter and the accuracy tolerance, a first number of times to sample the semantic analysis matrix to obtain the first subset of the rows of the semantic analysis matrix,

projecting each of the rows of the semantic analysis matrix onto a linear combination of the sampled first subset of the rows of the semantic analysis matrix;

determining a set of distances form the rows of the semantic analysis matrix to the linear combination based at least in part on the projecting;

determining, based at least in part on the target rank parameter, the accuracy tolerance, and the set of distances, a second number of times to sample the semantic analysis matrix to obtain a second subset of the rows of the semantic analysis matrix;

sampling the rows of the semantic analysis matrix the second number of times to obtain the second subset of the rows of the semantic analysis matrix; and

generating the lower-dimensional compressed matrix using at least a portion of a combined subset of the rows of the semantic analysis matrix, the combined subset containing the first subset and the second subset;

determining a first vector representation of a first document and a second vector representation of a second document;

determining a first lower-dimensional vector representation of the first document using the first vector representation and the lower-dimensional compressed matrix and a second lower-dimensional vector representation of the second document using the second vector representation and the lower-dimensional compressed matrix;

determining, using the first lower-dimensional vector representation and the second lower-dimensional vector representation, a similarity metric associated with the first document and the second document;

determining that the similarity metric satisfies a threshold value; and

classifying the first document and the second document in a same document cluster.

2. The computer-implemented method of claim 1 , wherein the lower-dimensional compressed matrix corresponds to a document subset of the set of documents that is representative of the set of documents within the accuracy tolerance.

3. The computer-implemented method of claim 1 , wherein determining the lower-dimensional compressed matrix further comprises determining an optimal subspace of the combined subset, wherein a number of dimensions of the optimal subspace is the target rank parameter.

4. A system for performing semantic document analysis, the system comprising:

at least one memory storing computer-executable instructions; and

at least one processor configured to access the at least one memory and execute the computer-executable instructions to:

determine a semantic analysis matrix corresponding to a set of documents, wherein each entry of the semantic analysis matrix indicates a number of times a respective term of a set of terms appears in a corresponding document in the set of documents;

determine a lower-dimensional compressed matrix from the semantic analysis matrix, wherein the at least one processor is configured to determine the lower-dimensional compressed matrix by executing the computer-executable instructions to:

determine a set of leverage scores corresponding to rows of the semantic analysis matrix; and

sample a first subset of the rows of the semantic analysis matrix based at least in part on a corresponding subset of leverage scores;

determine a target rank parameter;

determine an accuracy tolerance;

determine, based at least in part on the target rank parameter and the accuracy tolerance, a first number of times to sample the semantic analysis matrix to obtain the first subset of the rows of the semantic analysis matrix;

project each of the rows of the semantic analysis matrix onto a linear combination of the sampled first subset of the rows of the semantic analysis matrix;

determine a set of distances from the rows of the semantic analysis matrix to the linear combination based at least in part on the projecting;

determine, based at least in part on the target rank parameter, the accuracy tolerance, and the set of distances, a second number of times to sample the semantic analysis matrix to obtain a second subset of the rows of the semantic analysis matrix;

sample the rows of the semantic analysis matrix the second number of times to obtain the second subset of the rows of the semantic analysis matrix; and

generate the lower-dimensional compressed matrix using at least a portion of a combined subset of the rows of the semantic analysis matrix, the combined subset containing the first subset and the second subset;

determine a first vector representation of a first document and a second vector representation of a second document;

determine a first lower-dimensional vector representation of the first document using the first vector representation and the lower-dimensional compressed matrix and a second lower-dimensional vector representation of the second document using the second vector representation and the lower-dimensional compressed matrix;

determine, using the first lower-dimensional vector representation and the second lower-dimensional vector representation, a similarity metric associated with the first document and the second document;

determine that the similarity metric satisfies a threshold value; and

classify the first document and the second document in a same document cluster.

5. The system of claim 4 , wherein the lower-dimensional compressed matrix corresponds to a document subset of the set of documents that is representative of the set of documents within the accuracy tolerance.

6. The system of claim 4 , wherein the at least one processor is further configured to determine the lower-dimensional compressed matrix by executing the computer-executable instructions to determine an optimal subspace of the combined subset, wherein a number of dimensions of the optimal subspace is the target rank parameter.

7. A computer program product for performing semantic document analysis, the computer program product comprising a storage medium readable by a processing circuit, the storage medium storing instructions executable by the processing circuit to cause a method to be performed, the method comprising:

determining a semantic analysis matrix corresponding to a set of documents, wherein each entry of the semantic analysis matrix indicates a number of times a respective term of a set of terms appears in a corresponding document in the set of documents;

determining a lower-dimensional compressed matrix from the semantic analysis matrix, wherein determining the lower-dimensional compressed matrix comprises,

determining a set of leverage scores corresponding to rows of the semantic analysis matrix, and

sampling a first subset of the rows of the semantic analysis matrix based at least in part on a corresponding subset of the set of leverage scores;

determining a target rank parameter;

determining an accuracy tolerance;

determining, based at least in part on the target rank parameter and the accuracy tolerance, a first number of times to sample the semantic analysis matrix to obtain the first subset of the rows of the semantic analysis matrix;

projecting each of the rows of the semantic analysis matrix onto a linear combination of the sampled first subset of the rows of the semantic analysis matrix;

determining a set of distances from the rows of the semantic analysis matrix to the linear combination based at least in part on the projecting,

determining, based at least in part on the target rank parameter, the accuracy tolerance, and the set of distances, a second number of times to sample the semantic analysis matrix to obtain a second subset of the rows of the semantic analysis matrix,

sampling the rows of the semantic analysis matrix the second number of times to obtain the second subset of the rows of the semantic analysis matrix, and

generating the lower-dimensional compressed matrix using at least a portion of a combined subset of the rows of the semantic analysis matrix, the combined subset containing the first subset and the second subset;

determining a first vector representation of a first document and a second vector representation of a second document;

determining a first lower-dimensional vector representation of the first document using the first vector representation and the lower-dimensional compressed matrix and a second lower-dimensional vector representation of the second document using the second vector representation and the lower-dimensional compressed matrix;

determining, using the first lower-dimensional vector representation and the second lower-dimensional vector representation, a similarity metric associated with the first document and the second document;

determining that the similarity metric satisfies a threshold value; and

classifying the first document and the second document in a same document cluster.

8. The computer program product of claim 7 , wherein the lower-dimensional compressed matrix corresponds to a document subset of the set of documents that is representative of the set of documents within the accuracy tolerance.

9. The computer program product of claim 7 , wherein determining the lower-dimensional compressed matrix further comprises determining an optimal subspace of the combined subset, wherein a number of dimensions of the optimal subspace is the target rank parameter.