of our system that approximates the submodular objective function proposed by (Lin and Bilmes, 2011).7 As shown in the results, our best system8 which uses the hs dispersion function achieves a better ROUGE-1 F-score than all other systems.

(3) We also analyze the contributions of individual components of the new objective function towards summarization performance by selectively setting certain parameters to 0.

However, since individual components within our objective function are parametrized it is easy to tune them for a specific task or genre.

We start by describing a generic objective function that can be widely applied to several summarization scenarios.

This objective function is the sum of a monotone submodular coverage function and a non-submodular dispersion function.

We then describe a simple greedy algorithm for optimizing this objective function with provable approximation guarantees for three natural dispersion functions.

generate a graph and instantiate our summarization objective function with specific components that capture the desiderata of a given summarization task.

We model this property in our objective function as follows.

We then add this component to our objective function as w(S) = Zues Mu)-

The algorithm it-eratively adds to the current summary the element 3,- that has the largest ratio of the objective function gain to the additional cost, unless adding it violates the budget constraint.

After the loop, the algorithm compares Gi with the {3*} that has the largest value of the objective function among all subtrees that are under the budget, and it outputs the summary candidate with the largest value.

4.1 Objective Function

We designed our objective function by combining this relevance score with a penalty for redundancy and too-compressed sentences.

The behavior can be represented by a submodular objective function that reduces word scores depending on those already included in the summary.

Express the inference objective as a linear objective function ; and

For the grammatical error correction task, the variables in ILP are indicators of the corrections that a word needs, the objective function measures how grammatical the whole sentence is if some corrections are accepted, and the constraints guarantee that the corrections do not conflict with each other.

3.2 The Objective Function

(17) A new objective function combines the weights from both first and second order variables:

Variables of ILP are indicators of possible grammatical error corrections, the objective function aims to select the best set of corrections, and the constraints help to enforce a valid and grammatical output.

Wh611 noring the coupling between Vp, it can be solved by minimizing the objective function as follows:

Combining equations (1) and (2), we get the following objective function:

If we set a small value for Ap, the objective function behaves like the traditional NMF and the importance of data sparseness is emphasized; while a big value of Ap indicates Vp should be very closed to V1, and equation (3) aims to remove the noise introduced by statistical machine translation.

Gillick and Favre (Gillick and Favre, 2009) used bigrams as concepts, which are selected from a subset of the sentences, and their document frequency as the weight in the objective function .

where 0;, is an auxiliary variable we introduce that is equal to |nbflaef — :8 2(3) * 715,8 , and nbyef is a constant that can be dropped from the objective function .

To train this regression model using the given reference abstractive summaries, rather than trying to minimize the squared error as typically done, we propose a new objective function .

The objective function for training is thus to minimize the KL distance:

They used a modified objective function in order to consider whether the selected sentence is globally optimal.

This objective function can be optimized by the stochastic gradient method or other numerical optimization methods.

The squared-loss criterion1 is used to formulate the objective function .

Thus, the objective function can be optimized by L-BFGS-B (Zhu et al., 1997), a generic quasi-Newton gradient-based optimizer.

The first term in Equation (5) is the same as Equation (2), which is the traditional CRFs leam-ing objective function on the labeled data.

And third, the derived label information from the graph is smoothed into the model by optimizing a modified objective function .

Objective function : We aim to maximize: F : (pprosody + (pcoord + (Dneu

(Dneu : a Z wzged _ zj m Soft constraints (edge weights): The weights in the objective function are set as follows:

Objective function : We aim to maximize:

Hard constraints We add penalties to the objective function if the polarity of a pair of words is not consistent with its corresponding semantic relations.

Notice that dszjlr, d317,; satisfying above inequalities will be always of negative values, hence in order to maximize the objective function , the LP solver will try to minimize the absolute values of dsjj, dsgf, effectively pushing i and j toward the same polarity.

In that case, we can use a single parameter 04 to control both weights for different objective functions .

When 04 = 1 it is the unsupervised objective function in Formula (6).

Contrary, if 04 = 0, it is the projection objective function (Formula (7)) for projected instances.

We select a simple classifier objective function as the unsupervised objective function which is instinctively in accordance with the parsing objective:

Finally, the evolution of the objective function of the adapted K -means is modeled to automatically define the “best” number of clusters.

To assure convergence, an objective function Q is defined which decreases at each processing step.

The classical objective function is defined in Equation (1) where wk, is a cluster labeled k, xi 6 wk, is an object in the cluster, mm is the centroid of the cluster wk, and E(., is the Euclidean distance.

A direct consequence of the change in similarity measure is the definition of a new objective function Q53 to ensure convergence.

The objective function is defined as a linear combination of the potentials from different predictors with a parameter A to balance the contribution of two components: opinion entity identification and opinion relation extraction.

The objective function of ILP-W/O-ENTITY can be represented as

For ILP-W-SINGLE-RE, we simply remove the variables associated with one opinion relation in the objective function (1) and constraints.

The formulation of ILP-W/O-IMPLICIT—RE removes the variables associated with potential 7“,- in the objective function and corresponding constraints.

The gradient of the regularised objective function then becomes:

We learn the gradient using backpropagation through structure (Goller and Kuchler, 1996), and minimize the objective function using L-BFGS.

pred(l=l|v, 6) = Singid(Wlabel ’U + blabel) (9) Given our corpus of CCG parses with label pairs (N, l), the new objective function becomes:

The best parameters w* can be found by minimizing the following objective function:

The new objective function is changed as:

In the above objective function , we impose a user graph regularization term

The objective function can be transformed

Given the low-dimensional semantic representation of the test data, an objective function can be defined as follows:

The story boundaries which minimize the objective function 8 in Eq.

To implement this, we add a regularization term in the objective function of WTMF (equation 2) for each linked pairs ijl , 6233-2:

Therefore we approximate the objective function by treating the vector length |Q.,j as fixed values during the ALS iterations:

P and Q are optimized by minimize the objective function:

Let us rewrite the objective function as follows:

Note how this recovers the original objective function when matching variables are found.

This captures the positional information in the IBM models in a framework that admits exact parameter estimation inference, though the objective function is not concave: local maxima are a concern.

We will first briefly introduce single word vector representations and then describe the CVG objective function , tree scoring and inference.

The main objective function in Eq.

The objective function is not differentiable due to the hinge loss.

Input : relation instances R = {(indi,argi>}£f=1, generated abstracts A = {absfijib objective function f , cost function G

We employ the following objective function:

Algorithm 1 sequentially finds an abstract with the greatest ratio of objective function gain to length, and add it to the summary if the gain is nonnegative.

Our analysis shows that its objective function can be efficiently approximated using the negative empirical pointwise mutual information.

In this paper, we derive a new lower-bound approximate to the objective function used in the regularized compression algorithm.

The new objective function is written out as Equation (4).