sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: It is important to notice that the rotation planes of the Givens rotations are selected from the whole L-dimensional space whereas the objective function only uses information of correlations among the first R signal components u i . Successive application of Givens rotations Q µν leads to the final rotation matrix Q which is in the ideal case such that , but it is not clear if the final minimum is also the global one.
Sentence 2: in various simulations no local minima have been found.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To compare (2) and (3), note that our assumptions on the bounded entries of M ensures that: In particular, we see that the worst case bound for (3) is √ nm which is always lower than the worst case bound for (2): , our bound is only larger by a logarithmic term in m compared to [AM07].
Sentence 2: the algorithm proposed in [AM07] requires to store O(δmn) entries of M whereas SLA needs O(n) memory space.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, heads-up limit Texas hold'em, a relatively simple form of poker with 10 13 decision points, was essentially solved without decomposition [2].
Sentence 2: this approach cannot extend to larger games, such as heads-up no-limit Texas hold'em-the primary benchmark in imperfect-information game solving-which has 10 161 decision points [16].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A potential advantage of systems which fix the grammar and only perform parameter search is that they do compare complete grammars against each other, and are therefore able to detect which give rise to systematically compatible parses.
Sentence 2: although early work showed that small, artificial CFGs could be induced with the EM algorithm [12], studies with large natural language grammars have generally suggested that completely unsupervised EM over PCFGs is ineffective for grammar acquisition.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Deep structured models are widely used for tasks like semantic segmentation, where explicit correlations between variables provide important prior information which generally helps to reduce the data needs of deep nets.
Sentence 2: current deep structured models are restricted by oftentimes very local neighborhood structure, which cannot be increased for computational complexity reasons, and by the fact that the output configuration, or a representation thereof, cannot be transformed further.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Even if the image collection does not contain any labels, it has been shown to help tasks such as image completion and exploration [6,21], image colorization [22], and 3D surface layout estimation [5].
Sentence 2: as noted by several authors and illustrated in Figure 1, the major stumbling block of all the scene-matching approaches is that, despite the large quantities of data, for many types of images the quality of the matches is still not very good.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Several learning techniques naturally generalize to an arbitrary number of classes, such as density estimation, or logistic regression.
Sentence 2: when comparing the reported performance of these systems with the de-facto standard of using two-class techniques in combination with simple, fixed output codes to solve multi-class problems, they often lack in terms of performance, ease of optimization, and/or run-time behavior.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For instance, we notice some small overfitting for the DST-NN starting with hidden layer size 32, while the NN starts to show some small overfitting only at hidden layer size 128.
Sentence 2: the RNN does not show any sign of overfitting in 100 epochs of training, regardless the hidden layer size in the tested range, and the performance in training is somewhat equivalent to the performance in testing.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A common approach is to solve (1) for several values of λ, then use holdout data (or theoretical approaches, like AIC or SRM) to select a good value.
Sentence 2: if we view the regularized optimization problem as a family of problems, parameterized by the regularization parameter λ, it allows us to define the "path" of optimal solutions { β(λ) : 0 ≤ λ ≤ ∞}, which is a 1-dimensional curve through R p . Path following methods attempt to utilize the mathematical properties of this curve to devise efficient procedures for "following" it and generating the full set of regularized solutions with a (relatively) small computational cost.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: (3.1) A greedy policy π (i) can be obtained at each iteration i by ∀s : For any u > 0, it can be shown that if one can approximate u and run the above value iteration algorithm using these approximated values, then after Θ((1 ) iterations, the final iteration gives an value function that is u-optimal ([Ber13]).
Sentence 2: a u-optimal value function only yields a u/(1 − γ)-optimal greedy policy (in the worst case), even if (3.2) is precisely computed.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: From inspection of (3), it is clear that each of the elements of Y * is a linear combination of the sample averages Ȳ .
Sentence 2: a stronger statement can be made: Nt < ∞ for all t, then the MTA estimates {Y * t } given in (3) are a convex combination of the task sample averages { Ȳt }.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For two different kinds of hyperplane splits, if alternate split is invoked the same number of times in the tree, the difference in their worst-case guarantees for both the trees would again be governed by their worstcase quantization performance (β * ).
Sentence 2: for any fixed η, a harder question is whether one type of hyperplane partition violates the inlier condition more often than another type of partition, resulting in more alternate partitions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, then the dictionary size m affects the permissible radius of perturbation and the stability bound of the sparse coding stability.
Sentence 2: the factor of m does not appear in Theorem 2, and thus, the result is effective even for a large m. In addition, whereas ∥x∥ ≤ 1 is assumed in Mehta and Gray (2013), Theorem 2 does not assume that ∥x∥ ≤ 1 and clarifies the dependency for the norm ∥x∥.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A notable exception are approaches that try to infer the scene layout of indoor scenes in the form of 3D bounding boxes [13,22].
Sentence 2: these approaches can only cope with limited amounts of clutter (e.g., beds), and rely on the fact that indoor scenes satisfy very closely the manhattan world assumption, i.e., walls (and often objects) are aligned with the three dominant vanishing points.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Unfortunately, due to the r t term in the expression of Ẋ, the function (X, Z, t) → ( Ẋ, Ż) is not Lipschitz at t = 0, and we cannot directly apply the Cauchy-Lipschitz existence and uniqueness theorem.
Sentence 2: one can work around it by considering a sequence of approximating ODEs, similarly to the argument used in [28].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: They are characterized by a notion of diversity, as shown in Figure 1; points in the plane sampled from a DPP (center) are more spread out than those sampled independently (left).
Sentence 2: in many cases we would like to make use of the most likely configuration (MAP inference, right), which involves finding the largest principal minor of a positive semidefinite matrix.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Since ∇ t = − rt xt,rt , its ∞ -norm is large only when there is a stock with high reward r t,i while the learner puts a small weight x t,i on it.
Sentence 2: the reason that the weight x t,i is small is because the learner finds it performing poorly prior to round t, which means that the learner had better choices and actually should have performed better than stock i previously (that is, negative regret against stock i).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The computational complexity of COMBEXP is determined by the structure of Co(M) and COMB-EXP has O(T log(T )) time complexity due to the efficiency of interior-point methods.
Sentence 2: the computational complexity of COMBAND depends on the complexity of sampling from M. COMBAND may have a time complexity that is super-linear in T (see [16, page 217]).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: On one hand, if ω is always positive, then the feature space will be totally connected and every two features can interact with each other.
Sentence 2: if ω is chosen to be a Dirac-delta function δ ij , namely ω(X i , X j ) = 1 for i = j and is 0 otherwise, then the nonlocal block gets localized and will act like a simplified residual block.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For a fixed integer T , the four methods go through the dataset (in terms of multiplying A with column vectors) the same number of times.
Sentence 2: since LazySVD does not need block orthogonalization (as needed in PM and Krylov) and does not need a (T k)dimensional SVD computation in the end (as needed in Krylov), the running time of LazySVD is clearly much faster for a fixed value T . We therefore compare the performances of the four methods in terms of running time rather than T . We programmed the four algorithms using the same programming language with the same sparsematrix implementation.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Whenever tasks are somehow linearly dependent, even if the weight matrix W * is dense and s ≈ d, the rank s * can be small, thus guaranteeing a dramatic improvement over GL-FQI.
Sentence 2: learning a new representation comes at the cost of a worse dependency on d. In fact, the term log(d)/ √ T in GL-FQI, becomes d/T , implying that many more tasks are needed for FL-FQI to construct a suitable representation.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Trivially, we can appeal to this result in the linear optimization case to obtain a |D|T regret by setting K to be the size of D. However, the regret could have a very poor n dependence, as |D| could be exponential in n (or worse).
Sentence 2: note that in the full information case, we could appeal to the K-arm case to obtain O( T log |D|) regret, which in many cases is acceptable (such as when D is exponential in n).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: [3]) has emphasized that the utility of a VAE hinges on its ability to capture useful information in the latent representation z, complementary to that in powerful decoder networks p θ (x|z).
Sentence 2: renewed interest in programmatic representations in AI have sought modular, generalized, compositional representations for high-level concepts [4,5,6,7,8].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This knowledge may be incorporated by a weight w t which indicates the strength of a prior belief about whether the hypothesis is null or not-typically, a larger w t > 1 can be interpreted as a greater likelihood of being a non-null, indicating that the algorithm may be more aggressive in deciding whether to reject H t . Such p-value weighting was first suggested in the offline FDR context by [6], though earlier work employed it in the context of FWER control.
Sentence 2: as with penalty weights in the offline setting, offline prior weights are also usually rescaled to have unit mean, and then existing offline algorithms simply replace the p-value P t by the weighted p-value P t /w t . it is not obvious how to incorporate prior weights in the online setting.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If the target environment is arbitrary, or drastically different from the study environment, no causal relations can be learned and scientific progress will come to a standstill.
Sentence 2: the fact that scientific experimentation continues to provide useful information about our world suggests that certain environments share common characteristics and that, owed to these commonalities, causal claims would be valid even where experiments have never been performed.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In addition, let us also remark that this complexity can be further decreased if the number of distinct values of s y with non-zero probability mass is smaller than m. Moreover, the MEUF framework will not deliver an exact F-maximizer if the assumption of independence is violated.
Sentence 2: mEUF relies on a smaller number of parameters (m values representing marginal probabilities).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This seems much less restrictive.
Sentence 2: notice that if Pr(S 2 |S 1 ) ≥ 1 − , then Pr(S 2 |S 1 ) ≤ , which implies by definition of weak dependence that Pr(S 2 ) ≤ /α and therefore Pr(S 2 ) ≥ 1 − /α.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Another type of methods are based on MCMC (Friedman & Koller, 2003;Eaton & Murphy, 2007;Grzegorczyk & Husmeier, 2008;Niinimäki & Koivisto, 2013;Su & Borsuk, 2016) where graphs are sampled from the posterior distribution.
Sentence 2: there is no guarantee on the quality of the approximation in finite runs (may not mix well and converge in finite runs).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Similar empirical observations were made by the CryptoNets work [8].
Sentence 2: to ensure accuracy the values in the network must lie in the range [− p−1 2 , p−1 2 ]; this influences the choice of the prime p. we note that large primes increase the verifier and prover run-time because of the higher cost of performing modular additions and multiplications.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The first stage usually takes at least O(dN 2 ) time, which is prohibitive when N is large.
Sentence 2: several methods such as K-subspaces clustering [18], K-SVD [1] and online subspace clustering [25] do have linear time complexity O(rdN ) per iteration, however, there are no global or local convergence guarantees.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: On one hand, the collaborative nature makes SLFA capture deeper statistical structure hidden in the data set, compared to the matrix factorization problem with the Tikhonov regularization S 2 F or sparse coding with the sparsity-inducing regularization such as S 1 .
Sentence 2: sLFA encourages sparse interactions which is very different from previous works such as correlated topic Model [2] and latent Gaussian model [18], where the latent factors are densely related.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Each Σ (k) is of size d k × d k and the full covariance is of size k d k × k d k . Thus TGP is projected onto the entire tensor with respect to the latent factors U, including all zero and nonzero elements, rather than a (meaningful) subset of them.
Sentence 2: the real-world tensor data are usually extremely sparse, with a huge number of zero entries and a tiny portion of nonzero entries.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It is for their scalability that these models are frequently a method of choice for learning on large scale data sets (e.g., the implementation of linear SVM [13] has won the large scale learning challenge at ICML 2008 and KDD CUP 2010).
Sentence 2: as the set of linear hypotheses defined on a small or moderate number of input features is usually of low capacity these methods often learn inaccurate descriptions of target concepts.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If the dependency structure across a neural population is relatively homogeneous, then these copulas may be useful in that they can be estimated using far less data than required, e.g., for a full covariance matrix (which has O(n 2 ) parameters).
Sentence 2: if the dependencies within a population vary markedly for different pairs of neurons (as in the data set examined here), such copulas will lack the flexibility to capture the complicated dependencies within a full population.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In higher dimensions, it is possible to describe more complex joint distributions, with different marginals along different projections.
Sentence 2: although the model distribution for individual coefficients is similar to the fixed sparse priors of ICA and sparse coding, the model is fundamentally non-linear and might yield a different lower-level representation; the coefficients u are no longer mutually independent, and the optimal set of basis functions must account for this.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We note that the graph partitioning methods used in [3] are typically much faster than the time needed by SDP solvers (e.g., SeDuMi [15] and SDPT3) for comparable problem sizes.
Sentence 2: given the increasing interest in SDP in the last few years, we may expect the development of new algorithms, and faster/more efficient software tools.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The Bethe approach (and its generalizations) can be understood as consisting of two steps: (a) replacing the exact entropy −Φ * with a tree (or hypertree) approximation; and (b) replacing the marginal polytope MARG(G; φ) with constraint sets defined by tree (or hypertree) consistency conditions.
Sentence 2: since the (hyper)tree approximations used do not bound the exact entropy, the optimal values of Bethe-type variational problems do not provide a bound on the value of the log partition function Φ(θ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: More specifically, the iterations are as follows: x k+1 = arg min (5) If the prior R is appropriately chosen, such as x 1 , a closed-form solution for (3), i.e., a soft thresholding solution is naturally desirable.
Sentence 2: for some more complicated regularizations, e.g., a patch based prior [8], solving (3) is nontrivial, and may require iterative methods.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We conjecture that the integration time of the LINs in the first reservoir should ideally be long enough to capture the co-articulations between successive phonemes emerging from the dynamical constraints of the articulatory system.
Sentence 2: it has to remain short enough to avoid that information pointing to the presence of a short phoneme is too much blurred by the left phonetic context.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It is clear that in general no single member A ∈ A can contain all other members of A , since adding a null set (i.e., a set of measure zero) to A results in a larger set A ′ which is nevertheless still a member of A .
Sentence 2: we can find a member A * ∈ A which contains all but a null set of every other set in A .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This phenomenon has been observed by others as well-for example, Cutler and Zhao [7] demonstrate that their PERT classifier (which is similar to batch version of MF) achieves test accuracy comparable to Breiman-RF on many real world datasets.
Sentence 2: in the presence of irrelevant features, methods which choose splits independent of labels (MF, ERT-1) perform worse than Breiman-RF and ERT-k (but still better than ORF-Saffari) as indicated by the results on the dna dataset.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In practice, the prior is typically easy to sample from, but it is typically infeasible to generate exact posterior samples.
Sentence 2: in models where we can tractably sample from the joint distribution p(✓, z, y), we can generate exact posterior samples for simulated observations using the elementary fact that p(y) p(✓, z|y) = p(✓, z, y) = p(✓, z) p(y|✓, z).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Several researchers (see for instance [4] and [9]) have proposed statistical tests suited for 2-class classification tasks where the performance is measured in terms of the classification error (ratio of the number of errors and the number of examples), which enables the use of assumptions based on the fact that the error can be seen as a sum of random variables over the evaluation examples.
Sentence 2: various research domains prefer to measure the performance of their models using different indicators, such as the F 1 measure, used in information retrieval [11], described in Section 2.1.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A detailed comparison between their result and ours for such a structured formulation will be discussed in Section 4.
Sentence 2: under the assumption T . such an assumption holds only if HSGD selects at least O k 2 samples at the k-th iteration due to In this way, their result under WRS is of the same order as ours under WoRS.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For instance, a naive approach to make predictions about conjunctive queries would be the following: First, one would run an edge prediction model on all possible pairs of nodes, and-after obtaining these edge likelihoods-one would enumerate and score all candidate subgraphs that might satisfy a query.
Sentence 2: this naive enumeration approach could require computation time that is exponential in the number of existentially quantified (i.e., bound) variables in the query [12].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In particular, the O(1/ √ T ) and O(1/λT ) bounds apply for all T ∈ {1, 2, . . . , m}, namely up to one full pass over a random permutation of the entire data.
Sentence 2: our techniques are also applicable to a constant (> 1) number of passes, by randomly reshuffling the data after every pass.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For general kernels, the complexity of initialization is O(n 3 ), while the complexity of each iteration is O(n 2 ).
Sentence 2: using an incomplete Cholesky decomposition [5] makes all costs linear in n. After the convex optimization, we obtain a low-rank matrix M ∈ C k which is pointwise nonnegative with unit diagonal, of the form U U ⊤ where U ∈ R n×m .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: One class of latent-feature models for social networks has been developed by [9,10,11], who proposed real-valued vectors as latent representations of the entities in the network where depending on the model, either the distance, inner product, or weighted combination of the vectors corresponding to two entities affects the likelihood of there being a link between them.
Sentence 2: extending our high school student example, we might hope that instead of having arbitrary realvalued features (which are still useful for visualization), we would infer binary features where each feature could correspond to an attribute like "male" or "athlete."
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In analogy with LDA or DCA, if our data consisted of word counts, with f indexing words and n indexing documents, then the columns of W would describe topics and c f kn would denote the number of occurrences of word f stemming from topic k in document n. In our setting W is considered a free deterministic parameter to be estimated by maximum likelihood.
Sentence 2: h is treated as a nonnegative random latent variable over which we will integrate.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It is possible to define a group structure satisfying properties (i) and (ii) in which each edge of G * appears as a 2-node group; but then, a graph G generated from a group-induced model with a very large exponent α will contain all edges of G * with high probability, and a decentralized search algorithm will be able to follow these edges directly to construct a short path to the target.
Sentence 2: a lower bound for the case α > 1 can be obtained if we place one additional restriction on the group structure.
Relationship: contrasting