sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: DropMax (q = p) has an entropy regularizer to address this issue, but the model obtains suboptimal performance due to the inaccurate posterior estimation.
Sentence 2: the gating function of DropMax is optimally regularized to make a crude selection of candidate classes via the proposed variational inference framework, and shows consistent and significant improvements over the baselines across all datasets.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As stated in [7], in analogy to SVD, the rank of a tensor A can be defined as the minimum number r for decomposing the tensor into rank-one components as follows: where • denotes outer product, D ∈ R r×r×•••×r is an N -order diagonal tensor whose jth diagonal element is λ j , and The above decomposition model is called PARAFAC.
Sentence 2: this rank definition is a highly nonconvex and discontinuous function on the tensor space.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Intuitively, this suggests that the only favorable initialization schemes (from which convergence to a global maximum is possible) are those in which (1) all initialized centers fall in the neighborhood of exactly one cluster of true centers, (2) the number of centers initialized within each cluster of true centers exactly matches the number of true centers in that cluster.
Sentence 2: this observation alone only suffices to guarantee that the success probability is polynomially small in M . In order to demonstrate that the success probability is exponentially small in M , we need to further refine this construction.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In the specific, the posterior probability of y ∈ Y given a sample x ∈ X is inductively defined as: The same posterior probabilities with respect to a forest F ⊆ T can be obtained as in (2).
Sentence 2: each CS decision tree is trained in an iterative way and, similar to the case of standard decision trees, a decision about whether to branch new nodes or produce a leaf is taken based on a subset of the training samples Z ⊆ D. in contrast to the standard setting, the learning process depends on the order in which nodes of the tree are grown because split functions depend on t (<m) which in turn is affected by the node ordering.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Most known results on the exact solution of phase retrieval problems [8,29,10,38,36,35] pertain to generic random matrices, where the entries of A are independent subgaussian random variables.
Sentence 2: in practice it is almost impossible to design purely random measurement matrices: in many cases as we mentioned above, the measurement is much more structured -generated by passing a signal through a manually designed filter.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, for d = O( −2 log n) and Φ = poly(n), it improves over the "vanilla" bound by a factor of O(log n/ log log n) and is lossier than the bound of [6] by a factor of O(log log n).
Sentence 2: compared to the latter, our construction time is nearly linear in n. The comparison is summarized in Table 1.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Recently, Lee and Bresler [17] proposed an algorithm (ADMiRA) motivated by the orthogonal matching pursuit line of work in compressed sensing and show that for affine constraints with isometry constant δ 4k ≤ 0.04, their algorithm recovers the optimal solution.
Sentence 2: their method is not very efficient for large datasets and when the rank of the optimal solution is relatively large.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Researchers have previously attempted to cope with this issue by clustering or randomly downsampling the pool [19,20,21,22]; however, such strategies provide no guarantees as to the potential loss in active selection quality.
Sentence 2: when applying our approach for this task, we can consider orders of magnitude fewer points when making the next active label request, yet guarantee selections within a known error of the traditional exhaustive pool-based technique.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For such speech, only the second filter from 50-200 Hz will resolve the harmonic at 180 Hz.
Sentence 2: the first filter from 25-100 Hz will pass low frequency noise; the third filter from 100-400 Hz will pass the first and second harmonics at 180 Hz and 360 Hz, and the fourth filter from 200-800 Hz will pass the second through fourth harmonics at 360, 540, and 720 Hz.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Besides stand-alone matrix approximation methods, ensemble methods, e.g., DFC [15], LLORMA [12], WEMAREC [5], etc., and mixture models, e.g., MPMA [4], etc., have been proposed to improve the recommendation accuracy and/or scalability by weighing different base models across different users/items.
Sentence 2: the above methods do not consider using different ranks to derive different base models.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The standard approach for learning the parameters in presence of latent variables is to use an EM type algorithm [9].
Sentence 2: since the variational distribution is restricted to have the same system noise covariance (see Appendix A) as the true posterior, the EM algorithm would leave this covariance completely unchanged in the M step and cannot be used for learning this crucial parameter.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For all real-world datasets, the probability distributions D k are not readily available to the learner.
Sentence 2: corollary 4 extends the learning guarantees of our solution to the case where an estimate Dk is used in lieu of the ideal distribution D k . Thus, we used standard density estimation methods to derive an estimate Dk for each k ∈ [p].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In contrast, an algorithm resulting from the CWGD viewpoint (usually called potential booster) may not necessarily be a boosting algorithm in the probability approximately correct (PAC) learning sense.
Sentence 2: while it is rather difficult to construct a boosting algorithm based on the first view, the algorithmic frameworks, e.g., AnyBoost [4], resulting from the second viewpoint have proven to be particularly prolific when it comes to developing new boosting algorithms.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Approximations of the Ω-return could, in principle, replace the λ-return in the whole family of TD(λ) algorithms.
Sentence 2: using the Ω-return for TD(λ) raises several interesting questions that are beyond the scope of this initial work (e.g., is there a linear-time way to estimate the Ω-return?
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Let n denotes the input dimension, we show that width-(n + 4) ReLU networks can approximate any Lebesgue integrable function on n-dimensional space with respect to L 1 distance.
Sentence 2: except for a zero measure set, all Lebesgue integrable functions cannot be approximated by width-n ReLU networks, which demonstrate a phase transition.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: From the update in Equation ( 13), we observe that Gen-Oja is performing an Oja update but with a controlled Markovian noise.
Sentence 2: we would like to highlight that classical techniques in the study of stochastic approximation with Markovian noise (as the Poisson Equation [6,22]) were not enough to provide adequate control on the noise to show convergence.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We suspected that the choice of this distribution would have a significant impact on the accuracy of trained HBNNs, and performed a hyperparameter sweep by varying DistF romAvg in Algorithm 1 when training AlexNet on ImageNet as described in the following sections.
Sentence 2: much to our surprise, models trained with the same average bitwidth achieved nearly identical accuracies regardless of distribution.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We could naively generalize our approach for parallel bandits by observing for T /2 rounds, applying the truncated product factorization to write an expression for each P {Y |a} in terms of observational quantities and explicitly playing the actions for which the observational estimates were poor.
Sentence 2: it is no longer optimal to ignore the information we can learn about the reward for intervening on one variable from rounds in which we act on a different variable.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Now consider the 2q polynomials The count in Corollary 3 differentiates between positive, negative and zero signs.
Sentence 2: we are only concerned with the positivity of YiaXia (in the proof of Theorem 1) or of Xia − Tia (in the proof of Theorem 8), and do not need to differentiate between zero and negative values.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When CD was first introduced, algorithms did not differentiate between coordinates; each coordinate i 2 [d] was selected uniformly at random at each time step (see, e.g., [19,20]).
Sentence 2: recent works (see, e.g., [10,24,15]) have shown that exploiting the structure of the data and sampling the 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We train the LM in an adversarial way by minimizing the loss of LM of real sentences and maximizing the loss of transferred sentences.
Sentence 2: since the LM is a structured discriminator, we would hope that a language model trained on the real sentences will automatically assign high perplexity to sentences not in the target domain, hence negative samples from the generator may not be necessary.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In this paper, we prove lower bounds for the δ-related setting, which includes as a special case the commonly-studied setting of randomly partitioned data (in which case δ = O(1/ √ n)).
Sentence 2: our bounds do not apply for random partitioning, since they use δ-related constructions which do not correspond to randomly partitioned data.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The optimization problem with fixed γ leads to a new learning problem expressed as Note how the data fitting term elegantly and naturally encodes the transfer of source labels y s i through estimated labels of test samples with a weighting depending on the optimal transport matrix.
Sentence 2: this comes at the price of having a quadratic number N s N t of terms, which can be considered as computationally expensive.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The absolute location of orientation of the k-th map will be denoted by ξ k = ( x k y k θ k ) T ; here x and y are the Cartesian coordinates and θ is the orientation.
Sentence 2: from the scan matcher, we can retrieve relative displacement information of the form δ k,k−1 = ( ∆x k,k−1 ∆y k,k−1 ∆θ k,k−1 ) T which, if scan matching was errorfree, would enable us to recover absolute information via the following recursion (under the boundary condition ξ 0 = (0, 0, 0) T ) scan matching is not without errors.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Our final classifier has the weight vector w = u + v. We refer to the resulting algorithm as CODA (Cotraining for Domain Adaptation), which can be stated concisely with the following optimization problem: min w,u,v log e (u;L) + e (v;L) + γs(L, U, w) subject to: (1) The optimization is non-convex.
Sentence 2: as it is not particularly sensitive to initialization, we set u, v randomly and optimize with standard conjugate gradient descent 5 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Due to the large dataset size, CNN training is widely implemented using distributed methods such as dataparallel stochastic gradient descent (SGD) [9,10,11,12,13,14,15,16], where gradients computed by distributed nodes are summed after every iteration to update the CNN model of every node 2 .
Sentence 2: this gradient aggregation can dramatically hinder the expansion of such systems, for it incurs significant communication costs, and will become the system bottleneck when communication is slow.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, stimuli that occur at the same time and space are likely to be interrelated by a common cause.
Sentence 2: if the visual and auditory cues are incongruent, e.g., when dubbing one syllable onto a movie showing a person mouthing a different syllable, listeners typically report hearing a third syllable that represents a combination of what was seen and heard (McGurk effect, cf.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 1: Parameters: a, b, L > 0 2: Initialize: Receive input vector x t ∈ X 6: Adversarial setting: Suffer loss t (f t (x t )) 7: Receive subgradient s t ∈ partial t (f t (x t ))  This bound shows that, if the cumulative loss of the competitor is small, the regret can grow slower than √ T . It is worse than the regret bounds for smooth losses in [9,25] because when the cumulative loss of the competitor is equal to 0, the regret still grows as Õ f instead of being constant.
Sentence 2: the PiSTOL algorithm does not require the prior knowledge of the norm of the competitor function h, as all the ones in [9,25] do.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Since m(t), S(t), λ(t) and Ψ(t) are dependent on these parameters, one needs also to take the corresponding implicit derivatives into account.
Sentence 2: these implicit gradients vanish if the consistency constraints for the means (9) and the covariances (10), as well as the ones for the Lagrange multipliers (14-15), are satisfied.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Each residue in an amino acid sequence has two primary degrees of freedom: rotation around the C α -N bond, referred to as the φ torsion angle, and rotation around the C α -C bond, referred to as the ψ torsion angle.
Sentence 2: it is difficult to fit a response surface in the space of torsion angles because the energy function is highly irregular in this space; a slight change in a single torsion angle typically causes large global structural changes, which in turn cause large energy changes.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, for the squared loss, our problem can be cast as an instance of the trust region problem, which can be solved globally using the DCA algorithm of Tao and An [1998].
Sentence 2: note that problem (4) is not jointly convex in q and w. using the dual problem associated to w yields the following equivalent problem, it can be rewritten as follows:  6) is convex since D is a convex set, the first term in ( 6) is convex as a maximum of convex (linear) functions of r. This problem can be solved using standard descent methods, where, at each iteration, we solve a standard QP in ↵, which admits a closed-form solution.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It can be approximated as p(x)q(x) f (x) dx By performing a similar calculation for the denominator of ( 14), it can be shown to be approximately equal to 1 √ P1P1 .
Sentence 2: hence, the Laplacian pdf distance can be written as a risk function, given by Note that if P 1 = P 2 = 1 2 , then L = 2P e , where P e is the probability of error when assigning data points to the two clusters, that is Thus, in this case, minimizing L is equivalent to minimizing P e . in the case that P 1 = P 2 , (28) has an even more interesting interpretation.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As a reference, running ABLR NN and ABLR NN transfer in this setting took 1.2 ± 0.2 and 16.3 ± 1.6 seconds per BO iteration respectively.
Sentence 2: bOHAMIANN needed 1335.7 ± 236.5 seconds per BO iteration, which was about 80 times more expensive.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As may be seen, our analytic approximation results for the potential closely match the results obtained via simulations.
Sentence 2: the actual query complexity is significantly lower in practice than predicted through our analysis, due to the fact that we assumed a worst case scenario for pairwise queries, and set the number of comparisons to K. For the misclassification ratio, we observe that the general trend is as expected -the larger the number of clusters K, the larger the misclassification ratio.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We can now show that: The constraint Γ 0 is equivalent to (and can thus be replaced by) the following SPD constraint on a smaller matrix Γ: Since Γ is a principal submatrix of Γ (assuming at least one training label is equal to 1), lemma 2.1 indeed shows that Γ 0 implies Γ 0.
Sentence 2: note that by adding a column and corresponding row to Γ, the rank is not increased.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Some examples include [16,1,17] who explored the use of skip connections; [18] who pointed out the distinction of constructing a "deep" RNN from the view of the recurrent paths and the view of the input-to-hidden and hidden-to-output maps.
Sentence 2: they did not rigorously formalize the notion of "depth" and its implications in "deep" RNNs.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 13: By introducing additional artificial variables a, the KKT conditions can be transformed into the linear optimization min i a i subject to the constraints: (22) The only nonlinear constraint is the product α i β i = 0.
Sentence 2: this can be effectively implemented in the simplex procedure by modifying the selection of the pivot element to ensure that α i and β i are never both in the set of basic variables.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, one could replace the spectral norm by the Frobenius norm in Assumptions 2 and 4, which would simplify some of the proofs.
Sentence 2: using the Frobenius norm would be a much stronger assumption [18] Theorem 4 holds not just for simple graphs, but in the more general case when Â is a weighted graph (i.e.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: System Constraints So far, it is assumed in ( 6) that the classifier score, y t , that is generated after presenting the flash group, f t , is available instantaneously prior to selecting the flash group for the next time step, f s t+1 .
Sentence 2: this is typically not the case during real-time BCI implementation as there is a delay between presenting a stimulus and computing the corresponding classifier score due to processing a time window of EEG data, as illustrated in Figure 3.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: They also showed that the treereweighted max-product algorithm [9] can be used to solve the LP relaxation given by LOCAL(G), and that this succeeds in finding the MAP assignment for 339 of the 369 proteins in their data set.
Sentence 2: the optimal solution to the LP relaxation for the remaining 30 proteins, arguably the most difficult of the proteins, is fractional.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In [22], the authors note that the tensor trace norm • tr in equation ( 3) is a convex lower bound to R on the set The key insight behind this observation is summarized in Lemma 4, which we report in Appendix A.
Sentence 2: the authors of [22] leave open the question of whether the tensor trace norm is the convex envelope of R on the set G ∞ .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Under mild conditions [14], the ground-truth matrices in M • can be jointly triangularized, that is, there exists an orthogonal matrix U • that simultaneously renders all matrices where low(A) is the strictly lower triangular part of A, i.e., [low(A)] ij = A ij if i > j and 0 otherwise.
Sentence 2: when σ > 0 the observed matrices in M σ can only be approximately jointly triangularized, for instance by solving the following optimization problem where • denotes Frobenius norm and optimization is over the manifold O(d) of orthogonal matrices.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As can be seen, the firing rates are a linear function of vestibular input, in agreement with equations 17 and 18 of the Appendix.
Sentence 2: a linear firing-rate modulation by head velocity is not universal, for some parameters we have seen asymmetrically head-velocity tuning, with a preference for small head velocities (not shown).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Yet another approach to the multivariable case is to use KCDC measures as features in a multiclass classification problem for d-dimensional distributions.
Sentence 2: as noted in [13], this approach quickly becomes rather cumbersome as the number of labels grows super-exponentially in the number of variables due to the rapid increase of the number of DAGs that can be constructed from d variables.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Our contributions are threefold: 1.
Sentence 2: we propose an efficient algorithm that queries f at O(k log d) locations and recovers: (i) the active set S along with (ii) a uniform approximation to each φ l , l ∈ S. the existing error bounds in the statistics community [11,12,15] are in the much weaker L 2 (P) sense.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Our first result shows that this is indeed the case under three of the distance functions listed above, and a similar example works for a fourth.
Sentence 2: surprisingly, under the Kendall tau distance the MLE best alternatives are exactly the approval winners, and hence are polynomial-time computable, which stands in sharp contrast to the NP-hardness of computing them given rankings [5].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Such an asymptotic comparison is not very useful in the general case, since the only possible conclusion, if €(hDis,oo) < €(hGen,oo), is that logistic regression is the superior algorithm.
Sentence 2: as we saw previously, it is in the non-asymptotic case that the most interesting "two-regime" behavior is observed.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The E-step samples from the posterior and the M-step maximizes the average log joint probability of the samples and the data: where Q(θ) = 1 m m i=1 log p(y, u i |x, θ).
Sentence 2: there is a significant drawback to MCEM: If the number of samples m used in the M-step is too low then there is a risk of the hyperparameters overfitting to those samples.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The much smaller difference in MSE and 1-SSIM suggests that the L-CAE decoder does a better job at generalizing to unseen context-specific classes than the linear decoder.
Sentence 2: the images show that there are still important face-specific features (such as nose and eye definition) that are much better decoded by the L-CAE trained only on faces (Figure 5E).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In contrast, no tractable algorithm for the NSG learning has been proposed to date.
Sentence 2: it has been suggested that the NSG learning it better than the NPG learning due to the three differences [12]: (a) The NSG learning appropriately benefits from the concepts of Amari's NG learning, since the metric F s,a (θ) necessarily and sufficiently accounts for the probability distribution that the average reward depends on.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In this context, the QCP formulation tends to find a doubly stochastic matrix P which minimizes the "average error" between AP and PB.
Sentence 2: these spurious mismatching edges can be thought of as outliers, so we would want a metric promoting that AP and PB share the same active set (non zero entries representing edges), with the exception of some sparse entries.
Relationship: contrasting