A Solution to Plato ' s Problem : The Latent Semantic Analysis Theory of Acquisition , Induction , and Representation of Knowledge

computational method for its solution. A Psychological Description of LSA as a Theory of Learning, Memory, and Knowledge We give a more complete description of LSA as a mathematical model later when we use it to simulate lexical acquisition. However, an overall outline is necessary to understand a roughly equivalent psychological theory we wish to present first. The input to LSA is a matrix consisting of rows representing unitary event types by columns representing contexts in which instances of the event types appear. One example is a matrix of unique word types by many individual paragraphs in which the words are encountered, where a cell contains the number of times that a particular word type, say model, appears in a particular paragraph, say this one. After an initial transformation of the cell entries, this matrix is analyzed by a statistical technique called singular value decomposition (SVD) closely akin to factor analysis, which allows event types and individual contexts to be re-represented as points or vectors in a high dimensional abstract space (Golub, Lnk, & Overton, 1981 ). The final output is a representation from which one can calculate similarity measures between all pairs consisting of either event types or con-space (Golub, Lnk, & Overton, 1981 ). The final output is a representation from which one can calculate similarity measures between all pairs consisting of either event types or contexts (e.g., word-word, word-paragraph, or paragraph-paragraph similarities). Psychologically, the data that the model starts with are raw, first-order co-occurrence relations between stimuli and the local contexts or episodes in which they occur. The stimuli or event types may be thought of as unitary chunks of perception or memory. The first-order process by which initial pairwise associations are entered and transformed in LSA resembles classical conditioning in that it depends on contiguity or co-occurrence, but weights the result first nonlinearly with local occurrence frequency, then inversely with a function of the number of different contexts in which the particular component is encountered overall and the extent to which its occurrences are spread evenly over contexts. However, there are possibly important differences in the details as currently implemented; in particular, LSA associations are symmetrical; a context is associated with the individual events it contains by the same cell entry as the events are associated with the context. This would not be a necessary feature of the model; it would be possible to make the initial matrix asymmetrical, with a cell indicating the co-occurrence relation, for example, between a word and closely following words. Indeed, Lund and Burgess (in press; Lund, Burgess, & Atchley, 1995), and SchUtze (1992a, 1992b), have explored related models in which such data are the input. The first step of the LSA analysis is to transform each cell entry from the number of times that a word appeared in a particular context to the log of that frequency. This approximates the standard empirical growth functions of simple learning. The fact that this compressive function begins anew with each context also yields a kind of spacing effect; the association of A and B is greater if both appear in two different contexts than if they each appear twice in one context. In a second transformation, all cell entries for a given word are divided by the entropy for that word, Z p log p over all its contexts. Roughly speaking, this step accomplishes much the same thing as conditioning rules such as those described by Rescorla & Wagner (1972), in that it makes the primary association better represent the informative relation between the entities rather than the mere fact that they occurred together. Somewhat more formally, the inverse entropy measure estimates the degree to which observing the occurrence of a component specifies what context it is in; the larger the entropy of, say, a word, the less information its observation transmits about the places it has occurred, so the less usage-defined meaning it acquires, and conversely, the less the meaning of a particular context is determined by containing the word. It is interesting to note that automatic information retrieval methods (including LSA when used for the purpose) are greatly improved by transformations of this general form, the present one usually appearing to be the best (Harman, 1986). It does not seem far-fetched to believe that the necessary transform for good information retrieval, retrieval that brings back text corresponding to what a person has in mind when the person offers one or more query words, corresponds to the functional relations in basic associative processes. Anderson (1990) has drawn attention to the analogy between information retrieval in external systems and those in the human mind. It is not clear which way the relationship goes. Does information retrieval in automatic systems work best when it mimics the circumstances that make people think two things are related, or is there a general logic that tends to make them have similar forms? In automatic information retrieval the logic is usually assumed to be that idealized searchers have in mind exactly the same text as they would like the system to find and draw the words in 2 Although this exploratory process takes some advantage of chance, there is no reason why any number of dimensions should be much better than any other unless some mechanism like the one proposed is at work. In all cases, the model's remaining parameters were fitted only to its input (training) data and not to the criterion (generalization) test. THE LATENT SEMANTIC ANALYSIS THEORY OF KNOWLEDGE 217 their queries from that text (see Bookstein & Swanson, 1974). Then the system's challenge is to estimate the probability that each text in its store is the one that the searcher was thinking about. This characterization, then, comes full circle to the kind of communicative agreement model we outlined above: The sender issues a word chosen to express a meaning he or she has in mind, and the receiver tries to estimate the probability of each of the sender's possible messages. Gallistel (1990), has argued persuasively for the need to separate local conditioning or associative processes from global representation of knowledge. The LSA model expresses such a separation in a very clear and precise way. The initial matrix after transformation to log frequency divided by entropy represents the product of the local or pairwise processes? The subsequent analysis and dimensionality reduction takes all of the previously acquired local information and turns it into a unified representation of knowledge. Thus, the first processing step of the model, modulo its associational symmetry, is a rough approximation to conditioning or associative processes. However, the model's next steps, the singular value decomposition and dimensionality optimization, are not contained as such in any extant psychological theory of learning, although something of the kind may be hinted at in some modem discussions of conditioning and, on a smaller scale and differently interpreted, is often implicit and sometimes explicit in many neural net and spreading-activation architectures. This step converts the transformed associative data into a condensed representation. The condensed representation can be seen as achieving several things, although they are at heart the result of only one mechanism. First, the re-representation captures indirect, higher-order associations. That is, jf a particular stimulus, X, (e.g., a word) has been associated with some other stimulus, Y, by being frequently found in joint context (i.e., contiguity), and Y is associated with Z, then the condensation can cause X and Z to have similar representations. However, the strength of the indirect XZ association depends on much more than a combination of the strengths of XY and YZ. This is because the relation between X and Z also depends, in a wellspecified manner, on the relation of each of the stimuli, X, Y, and Z, to every other entity in the space. In the past, attempts to predict indirect associations by stepwise chaining rules have not been notably successful (see, e.g., Pollio, 1968;  1968). If associations correspond to distances in space, as supposed by LSA, stepwise chaining rules would not be expected to work well; if X is two units from Y and Y is two units from Z, all we know about the distance from X to Z is that it must be between zero and four. But with data about the distances between X, Y, Z, and other points, the estimate of XZ may be greatly improved by also knowing XY and YZ. An alternative view of LSA's effects is the one given earlier, the induction of a latent higher order similarity structure (thus its name) among representations of a large collection of events. Imagine, for example, that every time a stimulus (e.g., a word) is encountered, the distance between its representation and that of every other stimulus that occurs in close proximity to it is adjusted to be slightly smaller. The adjustment is then allowed to percolate through the whole previously constructed structure of relations, each point pulling on its neighbors until all settle into a compromise configuration (physical objects, weather systems, and Hopfield nets do this too; Hopfield, 1982). It is easy to see that the resulting relation between any two representations depends not only on direct experience with them but with everything else ever experienced. Although the current mathematical implementation of LSA does not work in this incremental way, its effects are much the same. The question, then, is whether such a mechanism, when combined with the statistics of experience, produces a faithful reflection of human knowledge. Finally, to anticipate what is developed later, the computational scheme used by LSA for combining and condensing local information into a common