The relaxation can be solved with a modified version of the Viterbi algorithm.

In order to compute the arg max of this optimization problem, we need to satisfy the constraints within the Viterbi algorithm.

We augment the Viterbi chart with a count vector d E D where D C leplll and d(i,j) is a count for the (i,j)’th constraint, i.e.

Figure 6 shows this constrained Viterbi relaxation approach.

The key algorithmic idea is to extend the Viterbi algorithm in order to consider possible adjacent links in the reverse direction.

Despite the new constraints, we can still compute L()\) in 0(I J (I + J time using a variant of the Viterbi algorithm.

Figure 5: Modified Viterbi algorithm for computing the adj a-cent agreement L()\).

Note that for both of these models we can solve the optimization problem exactly using the standard Viterbi algorithm for HMM decoding.

We then use the standard Viterbi update for computing the at variables, adding in the score of the y(i, j) necessary to satisfy the constraints.

Our decoder uses a simple variant of the Viterbi algorithm for solving a relaxed version of this model.

0 It is based on a novel relaxation for the model of DeNero and Macherey (2011), solvable with a variant of the Viterbi algorithm.

(2013) presented an approach to GPU parsing that sacrifices traditional sparsity in exchange for raw computational power, obtaining a system that can compute Viterbi parses for a high-quality grammar at about 164 sentences per second on a midrange GPU.

The resulting system is capable of computing over 404 Viterbi parses per second—more than a 2x speedup—on the same hardware.

Table 1: Performance numbers for computing Viterbi inside charts on 20,000 sentences of length $40 from the Penn Treebank.

The Viterbi inside loop for the grammar is inlined into a kernel.

(2013)’s system is able to compute Viterbi charts at 164 sentences per second, for sentences up to length 40.

The unpruned Viterbi computations in a fine grammar using the clustering method of Canny et al.

Together these kernels implement the Viterbi inside algorithm.

On a midrange GPU, their system can compute Viterbi derivations at 164 sentences per second on sentences of length 40 or less (see timing details below).

(2013) is that it only computes Viterbi parses.

The Viterbi algorithm is a reasonably effective method for parsing.

in linear time (in the number of tokens) following the generalized Viterbi algorithm.

A slightly modified version of Viterbi could also be used to find segmentations that are constrained to agree with some given motif boundaries, but can segment other parts of the sentence optimally under these constraints.

Here y’ = Decode(x(k),w) is the optimal Viterbi decoding using the current estimates of the weights.

The Viterbi alignment is determined using the Viterbi algorithm, similar to the FFNN-based model, where the model is sequentially applied from f1 to f J5.

5 Strictly speaking, we cannot apply the dynamic programming forward-backward algorithm (i.e., the Viterbi algorithm) due to the long alignment history of yi.

Thus, the Viterbi alignment is computed approximately using heuristic beam search.

Given a specific model, the best alignment ( Viterbi alignment) of the sentence pair (ff, (if) can be found as

a1 For example, the HMM model identifies the Viterbi alignment using the Viterbi algorithm.

The model finds the Viterbi alignment using the Viterbi algorithm, similar to the classic HMM model.

To enhance the accuracy of the pipeline, we add additional constraints in the Viterbi decoding of the first CRF.

Therefore, to improve its accuracy, we enforce additional commonsense constraints in its Viterbi decoding.

Similar to Mfgfi‘cf’, for , we also include additional constraints in the Viterbi decoding, by disallowing transitions between two ones, and disallowing the sequence to be all zeros if it contains all the remaining constituents in the document.

Our approach was to adopt a greedy bottom-up tree-building, with two linear-chain CRFs as local probabilistic models, and enforce reasonable constraints in the first CRF’s Viterbi decoding.

Specifically, in the Viterbi decoding of the first CRF, we include additional constraints elicited from common sense, to make more effective local decisions.

Unsupervised Grammar Induction Our first method for grammar induction is fully unsupervised Viterbi EM training of the Dependency Model with Valence (DMV) (Klein and Manning, 2004), with uniform initialization of the model parameters.

Constrained Grammar Induction Our second method, which we will refer to as DMV+C, induces grammar in a distantly supervised fashion by using a constrained parser in the E-step of Viterbi EM.

We contrast the DMV trained with Viterbi EM+uniform initialization (DMV), our constrained DMV (DMV+C), and our model’s MBR decoding of latent syntax (Marginalized) with other recent work: Spitkovsky et al.

(2010a) show that Viterbi (hard) EM training of the DMV with simple uniform initialization of the model parameters yields higher accuracy models than standard soft-EM

In Viterbi EM, the E—step finds the maximum likelihood corpus parse given the current model parameters.

The Viterbi algorithm ( Viterbi , 1967) is used for the decoding.

The shortest path algorithm for typo correction and Viterbi algorithm for PTC conversion are very closely related.

The idea of “statistical input method” was proposed by modeling PTC conversion as a hidden Markov model (HMM), and using Viterbi ( Viterbi , 1967) algorithm to decode the sequence.

To do this, we first take our phrasebook triples and construct sample string pairs <Epron, Pinyin-split> by pronouncing the phrasebook English with FST A, and by pronouncing the phrasebook Chinglish with FSTs D and C. Then we run the EM algorithm to learn FST B parameters (Table 3) and Viterbi alignments, such as:

First, we obtain Viterbi alignments using the phoneme-based model, e. g.:

EM learns values for parameters like P(nai te|night), plus Viterbi alignments such as: