To reduce the dimensionality of the feature space, and create dense representations for words, we perform SVD on F. We use the left singular vectors corresponding to the k largest singular values to compute a rank k approximation F, of F. We perform truncated SVD using SVDLIBCZ.

The number of singular vectors k selected in SVD , and the number of PLSR dimensions L are set respectively to 1000 and 50 for the remainder of the experiments described in the paper.

tent feature spaces separately for the source and the target domains using Singular Value Decomposition ( SVD ).

The SVD smoothing in the first step both reduces the data sparseness in distributional representations of individual words, as well as the dimensionality of the feature space, thereby enabling us to efficiently and accurately learn a prediction model using PLSR in the second step.

To evaluate the overall effect of the number of singular vectors k used in the SVD step, and the number of PLSR components L used in Algorithm 1, we conduct two experiments.

2000 SVD dimensions

Figure 3: The effect of SVD dimensions.

Linear predictors are then learnt to predict the occurrence of those pivots, and SVD is used to construct a lower dimensional representation in which a binary classifier is trained.

Specifically, we concatenate the textual and visual vectors and project them onto a lower dimensional latent space using SVD (Golub and Reinsch, 1970).

We furthermore report results obtained with Bruni et al.’s (2014) bimodal distributional model, which employs SVD to integrate co-occurrence-based textual representations with visual repre-

McRae 0.71 0.49 0.68 0.58 0.52 0.62 Attributes 0.58 0.61 0.68 0.46 0.56 0.58 SAE 0.65 0.60 0.70 0.52 0.60 0.64 SVD — — 0.67 — — 0.57 kCCA — — 0.57 — — 0.55 Bruni — — 0.52 — — 0.46 RNN-640 0.41 — — 0.34 — —

The automatically obtained textual and visual attribute vectors serve as input to SVD , kCCA, and our stacked autoencoder (SAE).

McRae 0.52 0.31 0.42 Attributes 0.35 0.37 0.33 SAE 0.36 0.35 0.43 SVD — — 0.39 kCCA — — 0.37 Bruni — — 0.34 RNN-640 0.32 — —

We also observe that simply concatenating textual and visual attributes (Attributes, T+V) performs competitively with SVD and better than kCCA.

SVD was applied to the document and dependency statistics and the top 1000 dimensions of each type were retained.

The SVD matrix for the original corpus data has correlation 0.4279 to the behavioral data, also below the 95 % confidence interval for all J NNSE models.

_ SVD (Text)

J NNSE(fMRI+Text) data performed on average 6% better than the best NNSE(Text), and exceeding even the original SVD corpus representations while maintaining interpretability.

Typically, VSMs are created by collecting word usage statistics from large amounts of text data and applying some dimensionality reduction technique like Singular Value Decomposition ( SVD ).

Singular Value Decomposition (SVD) SVD is the most widely used dimensionality reduction technique in distributional semantics (Turney and Pantel, 2010), and it has recently been exploited to combine visual and linguistic dimensions in the multimodal distributional semantic model of Bruni et al.

SVD smoothing is also a way to infer values of unseen dimensions in partially incomplete matrices, a technique that has been applied to the task of inferring word tags of unanno-tated images (Hare et al., 2008).

Assuming that the concept-representing rows of V8 and W8 are ordered in the same way, we apply the (k-truncated) SVD to the concatenated matrix [VSWS], such that [VSWS] = U192 kzgf is a k-rank approximation of the original matrix.6 The projection function is then:

k Model 1 2 3 5 10 20 Chance 1.1 2.2 3.3 5.5 11.0 22.0 SVD 1.9 5.0 8.1 14.5 29.0 48.6 CCA 3.0 6.9 10.7 17.9 31.7 51.7 lin 2.4 6.4 10.5 18.7 33.0 55.0 NN 3.9 6.6 10.6 21.9 37.9 58.2

For the SVD model, we set the number of dimensions to 300, a common choice in distributional semantics, coherent with the settings we used for the visual and linguistic spaces.

Surprisingly, the very simple lin method outperforms both CCA and SVD .

UZVT = SVD (W)

COMPUTESVD(W, k) UZVT = SVD (W)

By computing SVD (W) one obtains a compact representation of this column space in the form of an orthonormal basis 6T.

The computational bottleneck of CL-SCL is the SVD of the dense parameter matrix W. Here we follow Blitzer et al.

For the SVD computation the Lanczos algorithm provided by SVDLIBC is employed.4 We investigated an alternative approach to obtain a sparse W by directly enforcing sparse pivot predictors w; through Ll-regularization (Tsuruoka et al., 2009), but didn’t pursue this strategy due to unstable results.

Obviously the SVD is crucial to the success of CL-SCL if m is sufficiently large.

The method is based on applying singular value decomposition ( SVD ) to a matrix W representing the occurrence of words in documents.

SVD results in an approximation of W by the product of three matrices, one in which each word is represented as a low—dimensional vector, one in which each document is represented as a low dimensional vector, and a diagonal scaling matrix.

An important property of SVD is that the rows of US — which represents the words — behave similarly to the original rows of W, in the sense that the cosine similarity between two rows in US approximates the cosine similarity between the corre—

Moreover, algorithms which use operations such as the SVD have a limit to the corpora sizes they

The S-Space Package supports two common techniques: the Singular Value Decomposition ( SVD ) and randomized projections.

All matrix data structures are designed to seamlessly integrate with six SVD implementations for maximum portability, including SVDLIBJ1 , a Java port of SVDLIBCZ, a scalable sparse SVD library.

empirical accuracy on prior knowledge and maximum entropy by finding the top L eigenvectors of an extended covariance matrix2 via PCA or SVD .

However, despite of the potential problems of numerical stability, SVD requires massive computational space and 0(M3) computational time where M is feature dimension, which limits its usage for high-dimensional data (Trefethen et al., 1997).

This is mainly because SSH requires SVD to find the optimal hashing functions which is computational expensive.

Latent Semantic Analysis (LSA; Deerwester et al., 1990) We apply truncated SVD to a tf.idf weighted, cosine normalized count matrix, which

Latent Semantic Analysis (LSA), perhaps the best known VSM, explicitly learns semantic word vectors by applying singular value decomposition ( SVD ) to factor a term—document co-occurrence matrix.

It is typical to weight and normalize the matrix values prior to SVD .

The following lemma justifies the use of an SVD calculation as one method for finding values for U a and Va that satisfy condition 2:

These algorithms use spectral methods: that is, algorithms based on eigenvector decompositions of linear systems, in particular singular value decomposition ( SVD ).

The first step is to take an SVD of the training examples, followed by a projection of the training examples down to a low-dimensional space.

However, it is worth pointing out that the evaluated parameter subset encompasses settings (narrow context window, positive PMI, SVD reduction) that have been

For the count models, PMI is clearly the better weighting scheme, and SVD outperforms NMF as a dimensionality reduction technique.

PMI SVD 500 42 PMI SVD 400 46 PMI SVD 500 47 PMI SVD 300 50 PMI SVD 400 5 1 PMI NMF 300 5 2 PMI NMF 400 53 PMI SVD 300 53

We perform the singular value decomposition ( SVD ) (Golub and Kahan, 1965) for A at first, and then cut down each singular value.

Shrinkage step: UEVT = SVD (A), Z = U max(2 — TZMO) VT. end while end foreach

Shrinkage step: UEVT = SVD (A), Z = U max(2 — TZMO) VT.