CS276: Information Retrieval and Web Search

Pandu Nayak and Prabhakar Raghavan

Lecture 11: Text Classification;

Vector space classification

[Borrows slides from Ray Mooney]



Recap: Naïve Bayes classifiers

  • Classify based on prior weight of class and conditional parameter for what each word says:

  • Training is done by counting and dividing:

  • Don’t forget to smooth



The rest of text classification

  • Today:

    • Vector space methods for Text Classification

      • Vector space classification using centroids (Rocchio)

      • K Nearest Neighbors

      • Decision boundaries, linear and nonlinear classifiers

      • Dealing with more than 2 classes

  • Later in the course

    • More text classification

      • Support Vector Machines

      • Text-specific issues in classification



Recall: Vector Space Representation

  • Each document is a vector, one component for each term (= word).

  • Normally normalize vectors to unit length.

  • High-dimensional vector space:

    • Terms are axes

    • 10,000+ dimensions, or even 100,000+

    • Docs are vectors in this space

  • How can we do classification in this space?



Classification Using Vector Spaces

  • As before, the training set is a set of documents, each labeled with its class (e.g., topic)

  • In vector space classification, this set corresponds to a labeled set of points (or, equivalently, vectors) in the vector space

  • Premise 1: Documents in the same class form a contiguous region of space

  • Premise 2: Documents from different classes don’t overlap (much)

  • We define surfaces to delineate classes in the space



Documents in a Vector Space



Test Document of what class?



Test Document = Government



Aside: 2D/3D graphs can be misleading



Using Rocchio for text classification

  • Relevance feedback methods can be adapted for text categorization

    • As noted before, relevance feedback can be viewed as 2-class classification

      • Relevant vs. nonrelevant documents

  • Use standard tf-idf weighted vectors to represent text documents

  • For training documents in each category, compute a prototype vector by summing the vectors of the training documents in the category.

    • Prototype = centroid of members of class

  • Assign test documents to the category with the closest prototype vector based on cosine similarity.



Illustration of Rocchio Text Categorization



Definition of centroid

  • Where Dc is the set of all documents that belong to class c and v(d) is the vector space representation of d.

  • Note that centroid will in general not be a unit vector even when the inputs are unit vectors.



Rocchio Properties

  • Forms a simple generalization of the examples in each class (a prototype).

  • Prototype vector does not need to be averaged or otherwise normalized for length since cosine similarity is insensitive to vector length.

  • Classification is based on similarity to class prototypes.

  • Does not guarantee classifications are consistent with the given training data. 

                                                                                                                        Why not?



Rocchio Anomaly

  • Prototype models have problems with polymorphic (disjunctive) categories.


Rocchio classification

  • Rocchio forms a simple representation for each class: the centroid/prototype

  • Classification is based on similarity to / distance from the prototype/centroid

  • It does not guarantee that classifications are consistent with the given training data

  • It is little used outside text classification

    • It has been used quite effectively for text classification

    • But in general worse than Naïve Bayes

  • Again, cheap to train and test documents



kNN decision boundaries



Example: k=6 (6NN)



Nearest-Neighbor Learning Algorithm

  • Learning is just storing the representations of the training examples in D.

  • Testing instance x (under 1NN):

    • Compute similarity between x and all examples in D.

    • Assign x the category of the most similar example in D.

  • Does not explicitly compute a generalization or category prototypes.

  • Also called:

    • Case-based learning

    • Memory-based learning

    • Lazy learning

  • Rationale of kNN: contiguity hypothesis



kNN Is Close to Optimal

  • Cover and Hart (1967)

  • Asymptotically, the error rate of 1-nearest-neighbor classification is less than twice the Bayes rate [error rate of classifier knowing model that generated data]

  • In particular, asymptotic error rate is 0 if Bayes rate is 0.

  • Assume: query point coincides with a training point.

  • Both query point and training point contribute error → 2 times Bayes rate



k Nearest Neighbor

  • Using only the closest example (1NN) to determine the class is subject to errors due to:

    • A single atypical example.

    • Noise (i.e., an error) in the category label of a single training example.

  • More robust alternative is to find the k most-similar examples and return the majority category of these k examples.

  • Value of k is typically odd to avoid ties; 3 and 5 are most common.



k Nearest Neighbor Classification

  • kNN = k Nearest Neighbor

  • To classify a document d into class c:

  • Define k-neighborhood N as k nearest neighbors of d

  • Count number of documents i in N that belong to c

  • Estimate P(c|d) as i/k

  • Choose as class argmaxc P(c|d) [ = majority class]



Similarity Metrics

  • Nearest neighbor method depends on a similarity (or distance) metric.

  • Simplest for continuous m-dimensional instance space is Euclidean distance.

  • Simplest for m-dimensional binary instance space is Hamming distance (number of feature values that differ).

  • For text, cosine similarity of tf.idf weighted vectors is typically most effective.



Illustration of 3 Nearest Neighbor for Text Vector Space



3 Nearest Neighbor vs. Rocchio

  • Nearest Neighbor tends to handle polymorphic categories better than Rocchio/NB.



Nearest Neighbor with Inverted Index

  • Naively, finding nearest neighbors requires a linear search through |D| documents in collection

  • But determining k nearest neighbors is the same as determining the k best retrievals using the test document as a query to a database of training documents.

  • Use standard vector space inverted index methods to find the k nearest neighbors.

  • Testing Time: O(B|Vt|) where B is the average number of training documents in which a test-document word appears.

    • Typically B << |D|



kNN: Discussion

  • No feature selection necessary

  • Scales well with large number of classes

    • Don’t need to train n classifiers for n classes

  • Classes can influence each other

    • Small changes to one class can have ripple effect

  • Scores can be hard to convert to probabilities

  • No training necessary

    • Actually: perhaps not true. (Data editing, etc.)

  • May be expensive at test time

  • In most cases it’s more accurate than NB or Rocchio



kNN vs. Naive Bayes

  • Bias/Variance tradeoff

    • Variance ≈ Capacity

  • kNN has high variance and low bias.

    • Infinite memory

  • NB has low variance and high bias.

    • Decision surface has to be linear (hyperplane – see later)

  • Consider asking a botanist: Is an object a tree?

    • Too much capacity/variance, low bias

      • Botanist who memorizes

      • Will always say “no” to new object (e.g., different # of leaves)

    • Not enough capacity/variance, high bias

      • Lazy botanist

      • Says “yes” if the object is green

    • You want the middle ground

                                                                                (Example due to C. Burges)



Bias vs. variance: Choosing the correct model capacity



Linear classifiers and binary and multiclass classification

  • Consider 2 class problems

    • Deciding between two classes, perhaps, government and non-government

      • One-versus-rest classification

  • How do we define (and find) the separating surface?

  • How do we decide which region a test doc is in?



Separation by Hyperplanes

  • A strong high-bias assumption is linear separability:

    • in 2 dimensions, can separate classes by a line

    • in higher dimensions, need hyperplanes
  • Can find separating hyperplane by linear programming

    • (or can iteratively fit solution via perceptron):

    • separator can be expressed as ax + by = c



Linear programming / Perceptron

  • Find a,b,c, such that
  • ax + by > c for red points

  • ax + by < c for blue points.


Which Hyperplane?

  • In general, lots of possible solutions for a,b,c.



Which Hyperplane?

  • Lots of possible solutions for a,b,c.

  • Some methods find a separating hyperplane, but not the optimal one [according to some criterion of expected goodness]

    • E.g., perceptron

  • Most methods find an optimal separating hyperplane

  • Which points should influence optimality?

    • All points

      • Linear/logistic regression

      • Naïve Bayes

    • Only “difficult points” close to decision boundary

      • Support vector machines







Linear classifier: Example

  • Class: “interest” (as in interest rate)

  • Example features of a linear classifier


                        wi     ti                                              wi     ti

                     0.70 prime                                        −0.71 dlrs

                     0.67 rate                                           −0.35 world

                     0.63 interest                                     −0.33 sees

                     0.60 rates                                          −0.25 year

                     0.46 discount                                    −0.24 group

                     0.43 bundesbank                               −0.24 dlr

  • To classify, find dot product of feature vector and weights



Linear Classifiers

  • Many common text classifiers are linear classifiers

    • Naïve Bayes

    • Perceptron

    • Rocchio

    • Logistic regression

    • Support vector machines (with linear kernel)

    • Linear regression with threshold

  • Despite this similarity, noticeable performance differences

    • For separable problems, there is an infinite number of separating hyperplanes. Which one do you choose?

    • What to do for non-separable problems?

    • Different training methods pick different hyperplanes

  • Classifiers more powerful than linear often don’t perform better on text problems. Why?



Rocchio is a linear classifier



Two-class Rocchio as a linear classifier

  • Line or hyperplane defined by:

  • For Rocchio, set:

  • [Aside for ML/stats people: Rocchio classification is a simplification of the classic Fisher Linear Discriminant where you don’t model the variance (or assume it is spherical).]



Naive Bayes is a linear classifier

  • Two-class Naive Bayes. We compute:

  • Decide class C if the odds is greater than 1, i.e., if the log odds is greater than 0.

  • So decision boundary is hyperplane:



A nonlinear problem

  • A linear classifier like Naïve Bayes does badly on this task

  • kNN will do very well (assuming enough training data)



High Dimensional Data

  • Pictures like the one at right are absolutely misleading!

  • Documents are zero along almost all axes

  • Most document pairs are very far apart (i.e., not strictly orthogonal, but only share very common words and a few scattered others)

  • In classification terms: often document sets are separable, for most any classification

  • This is part of why linear classifiers are quite successful in this domain







More Than Two Classes

  • Any-of or multivalue classification

    • Classes are independent of each other.

    • A document can belong to 0, 1, or >1 classes.

    • Decompose into n binary problems

    • Quite common for documents

  • One-of or multinomial or polytomous classification

    • Classes are mutually exclusive.

    • Each document belongs to exactly one class

    • E.g., digit recognition is polytomous classification

      • Digits are mutually exclusive



Set of Binary Classifiers: Any of

  • Build a separator between each class and its complementary set (docs from all other classes).

  • Given test doc, evaluate it for membership in each class.

  • Apply decision criterion of classifiers independently

  • Done

    • Though maybe you could do better by considering dependencies between categories



Set of Binary Classifiers: One of

  • Build a separator between each class and its complementary set (docs from all other classes).

  • Given test doc, evaluate it for membership in each class.

  • Assign document to class with:



  • maximum score
  • maximum confidence

  • maximum probability



Summary: Representation ofText Categorization Attributes

  • Representations of text are usually very high dimensional (one feature for each word)

  • High-bias algorithms that prevent overfitting in high-dimensional space should generally work best*

  • For most text categorization tasks, there are many relevant features and many irrelevant ones

  • Methods that combine evidence from many or all features (e.g. naive Bayes, kNN) often tend to work better than ones that try to isolate just a few relevant features*

                                                    *Although the results are a bit more mixed than often thought



Which classifier do I use for a given text classification problem?

  • Is there a learning method that is optimal for all text classification problems?

  • No, because there is a tradeoff between bias and variance.

  • Factors to take into account:

    • How much training data is available?

    • How simple/complex is the problem? (linear vs. nonlinear decision boundary)

    • How noisy is the data?

    • How stable is the problem over time?

      • For an unstable problem, it’s better to use a simple and robust classifier.



Resources for today’s lecture

  • IIR 14

  • Fabrizio Sebastiani. Machine Learning in Automated Text Categorization. ACM Computing Surveys, 34(1):1-47, 2002.

  • Yiming Yang & Xin Liu, A re-examination of text categorization methods. Proceedings of SIGIR, 1999.

  • Trevor Hastie, Robert Tibshirani and Jerome Friedman, Elements of Statistical Learning: Data Mining, Inference and Prediction. Springer-Verlag, New York.

  • Open Calais: Automatic Semantic Tagging

    • Free (but they can keep your data), provided by Thompson/Reuters

  • Weka: A data mining software package that includes an implementation of many ML algorithms





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