Instead of evaluating the model on how well it predicts RTs for individual words or acoustic durations for individual words or segments, we can ask whether the model properly mirrors higher-level properties of the language system. More specifically, we can ask whether qualitative differences in lexical processing can be accounted for as resulting from differences in the distributional properties of cues and outcomes. If discriminative learning is on the right track, these differences should emerge straightforwardly from discriminative learning.
First consider the systematic differences in lexical processing that have been observed for English and Hebrew. For English, transposing letters (e.g., writingperhapsaspehraps) is relatively harmless.
Readers can reconstruct the intended meaning, and when reading for content, typos of this kind often escape notice. However, in Hebrew, letter transpositions severely disrupt reading (Frost, 2012).
To understand why, Baayen (2012) applied incremental NDL to English and Hebrew datasets of comparable size, using letter pairs as cues. Hebrew emerged as using a smaller set of bigram cues,
while using these cues more intensively, such that Hebrew bigram cues tended to have stronger weights to lexome outcomes compared to English. Interestingly, the disruption caused by transposing letters can be estimated by calculating the extent to which a word’s activation decreases when a pair of letters is transposed. Median disruption in Hebrew was almost 6.6 times that for English, which explains why Hebrew is more vulnerable to letter transpositions than English. Other systemic differences between Hebrew and English reading, such as the relative immunity of loanwords in Hebrew to letter transpositions, and the absence of facilitation from form priming in Hebrew, also follow straightforwardly from discriminative learning. Thus, discriminative learning helps clarify that indeed, as hypothesized by Frost (2012), the very different distributional properties of the Hebrew and English lexicons lie at the heart of the qualitative differences observed experiment-ally for lexical processing in these languages.
A second example of a system property that discriminative learning handles well is a double dissociation observed for English. Under impairment, English speakers with memory loss tend to have greater difficulties with irregular verbs, whereas speakers with phonological impairment tend to have more problems with regular verbs. Joanisse and Seidenberg (1999) used a connectionist network to model this dissociation, and observed that they had to selectively add noise to the semantic units of their model in order to get it to work properly. Interestingly, Joanisse and Seidenberg (1999) used one-hot encoding for their semantic representations, under the assumption that regular and irregular verbs do not differ systematically in meaning. However, irregular verbs have denser semantic neighborhoods than do regular verbs (Baayen and Moscoso del Prado Martín, 2005).
Heitmeier and Baayen (2021) therefore used LDL-EoL to model the double dissociation of regularity by impairment, as LDL makes it possible to work with corpus-based semantic vectors that reflect the differences in semantic density of regular and irregular verbs. Their simulation studies show that it is indeed the greater semantic density of English irregular verbs that renders them more fragile under semantic impairment.
A third example of using discriminative learning at the system level concerns second and third language acquisition. Chuang et al. (2020a) carried out a series of simulations with translation
equivalents from German, English, Mandarin, and Dutch, using both incremental LDL and LDL-EoL. As expected, the onset of L2 learning has a substantial influence on learning, as does the amount of L2 input. Another factor that emerged as important is the number of homophones in a language. Within the L1, homophones can be handled fairly well. However, when additional languages are learned, the homophones become more fragile and begin to suffer from intrusion, specifically if the translation equivalents of the other languages are not themselves homophonous.
The fragility of homophones fits well with the problems that L2 learners have with L2 words that have multiple senses that do not overlap with the senses of their L1 translation equivalents. Take English cut for example. One can cut with an axe, a knife, or scissors, whereas in Dutch and Mandarin the different cutting actions have to be expressed by three different verbs: hakken, knippen, andsnijdenin Dutch andkǎn, qiē, andjiǎnin Mandarin. L1 English speakers, when learning Dutch or Mandarin as L2, will thus have to learn to make more fine-grained decisions as to which verb to use as translation equivalent for Englishcut. From this perspective,cutactually has three meanings, and hence is a homophone. It straightforwardly follows that LDL predicts they are inevitably difficult for non-native speakers to learn.
5. General considerations
In computational linguistics, deep learning networks are the state-of-the-art. NDL/LDL contrasts with deep learning in several ways. First, there are no hidden layers. Second, the mathematics of multivariate multiple regression guarantees interpretational transparency. For the endstate of learning, convergence is to the global optimum. Third, LDL works well in sparse high dimensional spaces, whereas deep learning typically works with denser spaces of much lower dimensionality.
Since discriminative networks typically have many thousands of units, they implement ‘wide learning’.
Importantly, dimension reduction can lead to considerable reduction in LDL’s prediction accuracy.
For low-dimensional compressed spaces, deep learning is required, as mappings between compressed low-dimensional spaces typically are non-linear. In LDL, it is advisable to match the dimensionality of form vectors with that of the semantic vectors, for the mathematics to work best. As form matrices
using binary coding to specify the presence or absence of n-grams or n-phones are very sparse, networks mapping form onto meaning can be simplified by removing connections with weights very close to zero (see, e.g. Arnold et al., 2017; Milin et al., 2017b), with hardly any loss of accuracy.
Fourth, NDL and LDL hardly have any hyperparameters. For incremental learning, the learning rate is a free hyperparameter. For form vectors based on units such as triphones or trigrams, the dimensionality of the form space is determined by the number of different triphones in the data, and is not a free hyperparameter. And since the dimension of semantic vectors should match that of the form vectors, this is not a free hyperparameter either. Thus, how the model performs is determined by, firstly, the way form and meaning vectors are constructed (see Milin et al., 2017b, for discussion of the choice of representation for solving nonlinear classification with NDL), and secondly, by the distributional properties of the data. From this perspective, NDL/LDL provides a statistical tool that is driven almost exclusively by the data.
Breiman (2001) distinguished between two modeling cultures, the data modeling culture of statistics, and the algorithmic modeling culture of machine learning. The goal of algorithmic modeling is to obtain precise predictions, and if the model remains a black box or is theoretically uninterpretable (as is often the case for deep learning models as well as with recursive partitioning models) this is not an issue. The goal of data modeling, by contrast, is to obtain a statistical model that can generate the observed data. The resulting model provides the analyst with insight into what could be the mechanisms that give rise to the data. Although LDL can be seen as the simplest possible way of implementing machine learning for lexical learning, it is in spirit much closer to statistical data analysis. It is designed to assist the analyst to better understand the relations between sublexical distributional properties and text-derived distributional semantic properties.
Because LDL implementslinearmultivariate multiple regression, it inherits from the linear model the limitation that nonlinear functional relations (as addressed by, e.g., the generalized additive model, Wood, 2017) cannot be discovered. Here, there is room for considerable improvement.
An important property of the present formalization of Word and Paradigm Morphology is that morphological processing is not construed as involving parallel operations on symbols of form
and meaning. Figuring out the proper forms in production, or the proper meanings in compre-hension, is left to the network. Importantly, the network dynamically creates form and meaning representations, rather than retrieving them from some static lexical repository. Thus, NDL and LDL move away from classical compositional models in which stems and exponents have to be parsed out for comprehension or combined for production. However, the present approach is ana-lytical in the sense that semantic vectors for inflected words are obtained by summing the semantic vectors of pertinent lexomes.
It has been argued that phonemes, stems, and exponents have corresponding specialized neural areas in the brain (Bozic et al., 2010; Cibelli et al., 2015). Since NDL and LDL are statistical models of lexical processing, they make no claims about the topological organization of discrimination networks in neural tissue. However, it is mathematically possible to impose topological organization on form and semantic units in a 2-dimensional plane. As shown in Baayen et al. (2018) and Shafaei-Bajestan et al. (2020), phone-like and morph-like clusters can emerge in such 2-D maps. (As shown by Heitmeier and Baayen (2021), such maps make it possible to implement topologically localized lesions in simulation studies of aphasia.) Thus, NDL and LDL are not necessarily at odds with experimental evidence that is usually understood as supporting the neural reality of phonemes and morphemes. Instead of taking phonemes and morphemes to be the atomic units of a linguistic calculus, they can be understood as emergent properties of discriminative learning networks. Crucially, these low-level statistical networks provide much more precise predictions for the fine details of lexical processing than can be obtained with high-level abstract symbolic systems (for a discussion of NDL and statistical learning in language acquisition, see Baayen et al., 2016b).
However, since high-level symbolic descriptions are part of the cultural embedding of language, and have uncontestable pedagogical value, a full model of the mental lexicon will have to comprise not only low-level discriminative learning, but also high-level explicit symbolic learning that is grafted on top of low-level implicit learning.
Vectors and matrix multiplication
In what follows, we provide a brief conceptual introduction to vectors and matrices, using as example shopping lists and total expenses depending on supermarket. Consider Jack, who wants to buy two apples and three pairs, and Jill, who wants to buy four apples and two pairs. Their respective shopping lists can be represented as vectors, ordered sequences of numbers: (2,3) for Jack and (4,2)for Jill. We bring these two shopping list vectors together in a 2×2 matrixC.
C =
At the Edeka supermarket, apples coste2 and pears e3. At the Rewe supermarket, they coste3 ande2. We bundle this information in a second matrix,F.
F =
We can now calculate the total expenses at each of the supermarkets for both Jack and Jill. For Jack, buying 2 apples and 3 pears at Edeka will coste2×2 + 3×3 = 13. This combination of weighting numbers by price, and adding, is formalized by matrix multiplication.
S =CF =
The numbers going into the calculation of the expenses for Jack at Edeka are bolded: the row vector of Jack’s shopping list inCis paired with the column vector for Edeka inF, the elements of these vectors are multiplied pairwise, and then summed. The other three total expenses listed in S are
obtained in the same way. Jack is better off going to Rewe for his shopping, whereas Jill is better of going to Edeka.
When modeling comprehension with LDL, we knowC and we also know S, and we want to estimateF. WithF in hand, we can transform a ‘shopping list’ of triphones into a list of ‘semantic expenses’.
Links to digital materials
An R package for discriminative learning,WpmWithLdl, is available at http://www.sfs.uni-tuebingen.de/~hbaayen/publications/WpmWithLdl_1.0.tar.gz. An optimized implemen-tation of LDL in julia,JudiLing, is available athttps://megamindhenry.github.io/JudiLing.
jl/stable/.
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