I did not save the notebook I made during the first live coding session, but I attached to that WRI community post a modified version of the notebook I used for a Meetup workshop 6-7 months ago. (See for more details the GitHub MathematicaVsR project “Quantile Regression Workflows”.)
The notebook of the 2nd session is also attached. (I added a “References” section to it.)
In this document we describe the design and implementation of a (software programming) monad, [Wk1], for Latent Semantic Analysis workflows specification and execution. The design and implementation are done with Mathematica / Wolfram Language (WL).
What is Latent Semantic Analysis (LSA)? : A statistical method (or a technique) for finding relationships in natural language texts that is based on the so called Distributional hypothesis, [Wk2, Wk3]. (The Distributional hypothesis can be simply stated as “linguistic items with similar distributions have similar meanings”; for an insightful philosophical and scientific discussion see [MS1].) LSA can be seen as the application of Dimensionality reduction techniques over matrices derived with the Vector space model.
The goal of the monad design is to make the specification of LSA workflows (relatively) easy and straightforward by following a certain main scenario and specifying variations over that scenario.
The data for this document is obtained from WL’s repository and it is manipulated into a certain ready-to-utilize form (and uploaded to GitHub.)
The monadic programming design is used as a Software Design Pattern. The LSAMon monad can be also seen as a Domain Specific Language (DSL) for the specification and programming of machine learning classification workflows.
Here is an example of using the LSAMon monad over a collection of documents that consists of 233 US state of union speeches.
The table above is produced with the package “MonadicTracing.m”, [AAp2, AA1], and some of the explanations below also utilize that package.
As it was mentioned above the monad LSAMon can be seen as a DSL. Because of this the monad pipelines made with LSAMon are sometimes called “specifications”.
Remark: In this document with “term” we mean “a word, a word stem, or other type of token.”
Remark: LSA and Latent Semantic Indexing (LSI) are considered more or less to be synonyms. I think that “latent semantic analysis” sounds more universal and that “latent semantic indexing” as a name refers to a specific Information Retrieval technique. Below we refer to “LSI functions” like “IDF” and “TF-IDF” that are applied within the generic LSA workflow.
Contents description
The document has the following structure.
The sections “Package load” and “Data load” obtain the needed code and data.
The sections “Design consideration” and “Monad design” provide motivation and design decisions rationale.
The sections “LSAMon overview”, “Monad elements”, and “The utilization of SSparseMatrix objects” provide technical descriptions needed to utilize the LSAMon monad .
(Using a fair amount of examples.)
The section “Unit tests” describes the tests used in the development of the LSAMon monad.
(The random pipelines unit tests are especially interesting.)
The section “Future plans” outlines future directions of development.
The section “Implementation notes” just says that LSAMon’s development process and this document follow the ones of the classifications workflows monad ClCon, [AA6].
Remark: One can read only the sections “Introduction”, “Design consideration”, “Monad design”, and “LSAMon overview”. That set of sections provide a fairly good, programming language agnostic exposition of the substance and novel ideas of this document.
Package load
The following commands load the packages [AAp1–AAp7]:
In this section we load data that is used in the rest of the document. The text data was obtained through WL’s repository, transformed in a certain more convenient form, and uploaded to GitHub.
The text summarization and plots are done through LSAMon, which in turn uses the function RecordsSummary from the package “MathematicaForPredictionUtilities.m”, [AAp7].
In some of the examples below we want to explicitly specify the stop words. Here are stop words derived using the built-in functions DictionaryLookup and DeleteStopwords.
Here is a quote from [Wk1] that fairly well describes why we choose to make a classification workflow monad and hints on the desired properties of such a monad.
[…] The monad represents computations with a sequential structure: a monad defines what it means to chain operations together. This enables the programmer to build pipelines that process data in a series of steps (i.e. a series of actions applied to the data), in which each action is decorated with the additional processing rules provided by the monad. […]
Monads allow a programming style where programs are written by putting together highly composable parts, combining in flexible ways the possible actions that can work on a particular type of data. […]
Remark: Note that quote from [Wk1] refers to chained monadic operations as “pipelines”. We use the terms “monad pipeline” and “pipeline” below.
Monad design
The monad we consider is designed to speed-up the programming of LSA workflows outlined in the previous section. The monad is named LSAMon for “Latent Semantic Analysis** Mon**ad”.
We want to be able to construct monad pipelines of the general form:
LSAMon-Monad-Design-formula-1
LSAMon is based on the State monad, [Wk1, AA1], so the monad pipeline form (1) has the following more specific form:
LSAMon-Monad-Design-formula-2
This means that some monad operations will not just change the pipeline value but they will also change the pipeline context.
In the monad pipelines of LSAMon we store different objects in the contexts for at least one of the following two reasons.
The object will be needed later on in the pipeline, or
The object is (relatively) hard to compute.
Such objects are document-term matrix, Dimensionality reduction factors and the related topics.
Let us list the desired properties of the monad.
Rapid specification of non-trivial LSA workflows.
The text data supplied to the monad can be: (i) a list of strings, or (ii) an association with string values.
The monad uses the Linear vector space model .
The document-term frequency matrix can be created after removing stop words and/or word stemming.
It is easy to specify and apply different LSI weight functions. (Like “IDF” or “GFIDF”.)
The monad can do dimension reduction with SVD and NNMF and corresponding matrix factors are retrievable with monad functions.
Documents (or query strings) external to the monad are easily mapped into monad’s Linear vector space of terms and Linear vector space of topics.
The monad allows of cursory examination and summarization of the data.
The pipeline values can be of different types. (Most monad functions modify the pipeline value; some modify the context; some just echo results.)
It is easy to obtain the pipeline value, context, and different context objects for manipulation outside of the monad.
It is easy to tabulate extracted topics and related statistical thesauri.
The LSAMon components and their interactions are fairly simple.
The main LSAMon operations implicitly put in the context or utilize from the context the following objects:
document-term matrix,
the factors obtained by matrix factorization algorithms,
LSI weight functions specifications,
extracted topics.
Note the that the monadic set of types of LSAMon pipeline values is fairly heterogenous and certain awareness of “the current pipeline value” is assumed when composing LSAMon pipelines.
Obviously, we can put in the context any object through the generic operations of the State monad of the package “StateMonadGenerator.m”, [AAp1].
LSAMon overview
When using a monad we lift certain data into the “monad space”, using monad’s operations we navigate computations in that space, and at some point we take results from it.
With the approach taken in this document the “lifting” into the LSAMon monad is done with the function LSAMonUnit. Results from the monad can be obtained with the functions LSAMonTakeValue, LSAMonContext, or with the other LSAMon functions with the prefix “LSAMonTake” (see below.)
Here is a corresponding diagram of a generic computation with the LSAMon monad:
LSAMon-pipeline
Remark: It is a good idea to compare the diagram with formulas (1) and (2).
Let us examine a concrete LSAMon pipeline that corresponds to the diagram above. In the following table each pipeline operation is combined together with a short explanation and the context keys after its execution.
Here is the output of the pipeline:
The LSAMon functions are separated into four groups:
operations,
setters and droppers,
takers,
State Monad generic functions.
Monad functions interaction with the pipeline value and context
An overview of the those functions is given in the tables in next two sub-sections. The next section, “Monad elements”, gives details and examples for the usage of the LSAMon operations.
In this section we show that LSAMon has all of the properties listed in the previous section.
The monad head
The monad head is LSAMon. Anything wrapped in LSAMon can serve as monad’s pipeline value. It is better though to use the constructor LSAMonUnit. (Which adheres to the definition in [Wk1].)
The fundamental model of LSAMon is the so called Vector space model (or the closely related Bag-of-words model.) The document-term matrix is a linear vector space representation of the documents collection. That representation is further used in LSAMon to find topics and statistical thesauri.
Here is an example of ad hoc construction of a document-term matrix using a couple of paragraphs from “Hamlet”.
When we construct the document-term matrix we (often) want to stem the words and (almost always) want to remove stop words. LSAMon’s function LSAMonMakeDocumentTermMatrix makes the document-term matrix and takes specifications for stemming and stop words.
After making the document-term matrix we will most likely apply LSI weight functions, [Wk2], like “GFIDF” and “TF-IDF”. (This follows the “standard” approach used in search engines for calculating weights for document-term matrices; see [MB1].)
Frequency matrix
We use the following definition of the frequency document-term matrix F.
Each entry fij of the matrix F is the number of occurrences of the term j in the document i.
Weights
Each entry of the weighted document-term matrix M derived from the frequency document-term matrix F is expressed with the formula
where gj – global term weight; lij – local term weight; di – normalization weight.
Various formulas exist for these weights and one of the challenges is to find the right combination of them when using different document collections.
Here is a table of weight functions formulas.
LSAMon-LSI-weight-functions-table
Computation specifications
LSAMon function LSAMonApplyTermWeightFunctions delegates the LSI weight functions application to the package “DocumentTermMatrixConstruction.m”, [AAp4].
Here we are summaries of the non-zero values of the weighted document-term matrix derived with different combinations of global, local, and normalization weight functions.
Streamlining topic extraction is one of the main reasons LSAMon was implemented. The topic extraction correspond to the so called “syntagmatic” relationships between the terms, [MS1].
Theoretical outline
The original weighed document-term matrix M is decomposed into the matrix factors W and H.
M ≈ W.H, W ∈ ℝm × k, H ∈ ℝk × n.
The i-th row of M is expressed with the i-th row of W multiplied by H.
The rows of H are the topics. SVD produces orthogonal topics; NNMF does not.
The i-the document of the collection corresponds to the i-th row W. Finding the Nearest Neighbors (NN’s) of the i-th document using the rows similarity of the matrix W gives document NN’s through topic similarity.
The terms correspond to the columns of H. Finding NN’s based on similarities of H’s columns produces statistical thesaurus entries.
The term groups provided by H’s rows correspond to “syntagmatic” relationships. Using similarities of H’s columns we can produce term clusters that correspond to “paradigmatic” relationships.
Computation specifications
Here is an example using the play “Hamlet” in which we specify additional stop words.
One of the most natural operations is to find the representation of an arbitrary document (or sentence or a list of words) in monad’s Linear vector space of terms. This is done with the function LSAMonRepresentByTerms.
Here is an example in which a sentence is represented as a one-row matrix (in that space.)
obj =
lsaHamlet⟹
LSAMonRepresentByTerms["Hamlet, Prince of Denmark killed the king."]⟹
LSAMonEchoValue;
Here we display only the non-zero columns of that matrix.
obj⟹
LSAMonEchoFunctionValue[MatrixForm[Part[#, All, Keys[Select[SSparseMatrix`ColumnSumsAssociation[#], # > 0& ]]]]& ];
Transformation steps
Assume that LSAMonRepresentByTerms is given a list of sentences. Then that function performs the following steps.
1. The sentence is split into a list of words.
2. If monad’s document-term matrix was made by removing stop words the same stop words are removed from the list of words.
3. If monad’s document-term matrix was made by stemming the same stemming rules are applied to the list of words.
4. The LSI global weights and the LSI local weight and normalizer functions are applied to sentence’s contingency matrix.
Equivalent representation
Let us look convince ourselves that documents used in the monad to built the weighted document-term matrix have the same representation as the corresponding rows of that matrix.
Here is an association of documents from monad’s document collection.
inds = {6, 10};
queries = Part[lsaHamlet⟹LSAMonTakeDocuments, inds];
queries
(* <|"id.0006" -> "Getrude, Queen of Denmark, mother to Hamlet. Ophelia, daughter to Polonius.",
"id.0010" -> "ACT I. Scene I. Elsinore. A platform before the Castle."|> *)
lsaHamlet⟹
LSAMonRepresentByTerms[queries]⟹
LSAMonEchoFunctionValue[MatrixForm[Part[#, All, Keys[Select[SSparseMatrix`ColumnSumsAssociation[#], # > 0& ]]]]& ];
Another natural operation is to find the representation of an arbitrary document (or a list of words) in monad’s Linear vector space of topics. This is done with the function LSAMonRepresentByTopics.
Here is an example.
inds = {6, 10};
queries = Part[lsaHamlet⟹LSAMonTakeDocuments, inds];
Short /@ queries
(* <|"id.0006" -> "Getrude, Queen of Denmark, mother to Hamlet. Ophelia, daughter to Polonius.",
"id.0010" -> "ACT I. Scene I. Elsinore. A platform before the Castle."|> *)
lsaHamlet⟹
LSAMonRepresentByTopics[queries]⟹
LSAMonEchoFunctionValue[MatrixForm[Part[#, All, Keys[Select[SSparseMatrix`ColumnSumsAssociation[#], # > 0& ]]]]& ];
In order to clarify what the function LSAMonRepresentByTopics is doing let us go through the formulas it is based on.
The original weighed document-term matrix M is decomposed into the matrix factors W and H.
M ≈ W.H, W ∈ ℝm × k, H ∈ ℝk × n
The i-th row of M is expressed with the i-th row of W multiplied by H.
mi ≈ wi.H.
For a query vector q0 ∈ ℝm we want to find its topics representation vector x ∈ ℝk:
q0 ≈ x.H.
Denote with H( − 1) the inverse or pseudo-inverse matrix of H. We have:
q0.H( − 1) ≈ (x.H).H( − 1) = x.(H.H( − 1)) = x.I,
x ∈ ℝk, H( − 1) ∈ ℝn × k, I ∈ ℝk × k.
In LSAMon for SVD H( − 1) = HT; for NNMF H( − 1) is the pseudo-inverse of H.
The vector x obtained with LSAMonRepresentByTopics.
Tags representation
Sometimes we want to find the topics representation of tags associated with monad’s documents and the tag-document associations are one-to-many. See [AA3].
Let us consider a concrete example – we want to find what topics correspond to the different presidents in the collection of State of Union speeches.
Here we find the document tags (president names in this case.)
There are several algorithms we can apply for finding the most important documents in the collection. LSAMon utilizes two types algorithms: (1) graph centrality measures based, and (2) matrix factorization based. With certain graph centrality measures the two algorithms are equivalent. In this sub-section we demonstrate the matrix factorization algorithm (that uses SVD.)
Definition: The most important sentences have the most important words and the most important words are in the most important sentences.
That definition can be used to derive an iterations-based model that can be expressed with SVD or eigenvector finding algorithms, [LE1].
Here we pick an important part of the play “Hamlet”.
focusText =
First@Pick[textHamlet, StringMatchQ[textHamlet, ___ ~~ "to be" ~~ __ ~~ "or not to be" ~~ ___, IgnoreCase -> True]];
Short[focusText]
(* "Ham. To be, or not to be- that is the question: Whether 'tis ....y.
O, woe is me T' have seen what I have seen, see what I see!" *)
LSAMonUnit[StringSplit[ToLowerCase[focusText], {",", ".", ";", "!", "?"}]]⟹
LSAMonMakeDocumentTermMatrix["StemmingRules" -> {}, "StopWords" -> Automatic]⟹
LSAMonApplyTermWeightFunctions⟹
LSAMonFindMostImportantDocuments[3]⟹
LSAMonEchoFunctionValue[GridTableForm];
LSAMon-Find-most-important-documents-table
Setters, droppers, and takers
The values from the monad context can be set, obtained, or dropped with the corresponding “setter”, “dropper”, and “taker” functions as summarized in a previous section.
For example:
p = LSAMonUnit[textHamlet]⟹LSAMonMakeDocumentTermMatrix[Automatic, Automatic];
p⟹LSAMonTakeMatrix
If other values are put in the context they can be obtained through the (generic) function LSAMonTakeContext, [AAp1]:
Short@(p⟹QRMonTakeContext)["documents"]
(* <|"id.0001" -> "1604", "id.0002" -> "THE TRAGEDY OF HAMLET, PRINCE OF DENMARK", <<220>>, "id.0223" -> "THE END"|> *)
Another generic function from [AAp1] is LSAMonTakeValue (used many times above.)
Here is an example of the “data dropper” LSAMonDropDocuments:
(The “droppers” simply use the state monad function LSAMonDropFromContext, [AAp1]. For example, LSAMonDropDocuments is equivalent to LSAMonDropFromContext[“documents”].)
The utilization of SSparseMatrix objects
The LSAMon monad heavily relies on SSparseMatrix objects, [AAp6, AA5], for internal representation of data and computation results.
A SSparseMatrix object is a matrix with named rows and columns.
In some cases we want to show only columns of the data or computation results matrices that have non-zero elements.
Here is an example (similar to other examples in the previous section.)
lsaHamlet⟹
LSAMonRepresentByTerms[{"this country is rotten",
"where is my sword my lord",
"poison in the ear should be in the play"}]⟹
LSAMonEchoFunctionValue[ MatrixForm[#1[[All, Keys[Select[ColumnSumsAssociation[#1], #1 > 0 &]]]]] &];
In the pipeline code above: (i) from the list of queries a representation matrix is made, (ii) that matrix is assigned to the pipeline value, (iii) in the pipeline echo value function the non-zero columns are selected with by using the keys of the non-zero elements of the association obtained with ColumnSumsAssociation.
Similarities based on representation by terms
Here is way to compute the similarity matrix of different sets of documents that are not required to be in monad’s document collection.
Similarly to weighted Boolean similarities matrix computation above we can compute a similarity matrix using the topics representations. Note that an additional normalization steps is required.
Note the differences with the weighted Boolean similarity matrix in the previous sub-section – the similarities that are less than 1 are noticeably larger.
Unit tests
The development of LSAMon was done with two types of unit tests: (i) directly specified tests, [AAp7], and (ii) tests based on randomly generated pipelines, [AA8].
The unit test package should be further extended in order to provide better coverage of the functionalities and illustrate – and postulate – pipeline behavior.
Since the monad LSAMon is a DSL it is natural to test it with a large number of randomly generated “sentences” of that DSL. For the LSAMon DSL the sentences are LSAMon pipelines. The package “MonadicLatentSemanticAnalysisRandomPipelinesUnitTests.m”, [AAp9], has functions for generation of LSAMon random pipelines and running them as verification tests. A short example follows.
AbsoluteTiming[
res = TestRunLSAMonPipelines[pipelines, "Echo" -> False];
]
From the test report results we see that a dozen tests failed with messages, all of the rest passed.
rpTRObj = TestReport[res]
(The message failures, of course, have to be examined – some bugs were found in that way. Currently the actual test messages are expected.)
Future plans
Dimension reduction extensions
It would be nice to extend the Dimension reduction functionalities of LSAMon to include other algorithms like Independent Component Analysis (ICA), [Wk5]. Ideally with LSAMon we can do comparisons between SVD, NNMF, and ICA like the image de-nosing based comparison explained in [AA8].
Another direction is to utilize Neural Networks for the topic extraction and making of statistical thesauri.
Conversational agent
Since LSAMon is a DSL it can be relatively easily interfaced with a natural language interface.
Here is an example of natural language commands parsed into LSA code using the package [AAp13].
The implementation methodology of the LSAMon monad packages [AAp3, AAp9] followed the methodology created for the ClCon monad package [AAp10, AA6]. Similarly, this document closely follows the structure and exposition of the `ClCon monad document “A monad for classification workflows”, [AA6].
A lot of the functionalities and signatures of LSAMon were designed and programed through considerations of natural language commands specifications given to a specialized conversational agent.
Mathematica for prediction algorithms 