sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: To overcome this √ d gradient evaluation barrier, we therefore need to control the change in the Hessian with respect to a norm which does not grow as quickly with the dimension as the Euclidean norm for a random N (0, I d ) momentum vector.
Sentence 2: in this paper, we show that the conjecture of [11], which says that HMC requires O * (d 1 4 ) gradient evaluations, is true when sampling from strongly log-concave targets satisfying weak regularity properties associated with the input data.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We analyze the correctness and convergence time of A ⇤ Sampling and demonstrate empirically that it makes more efficient use of bound and likelihood evaluations than the most closely related adaptive rejection sampling-based algorithms.
Sentence 2: the distribution over the number of iterations is distributed as a Geometric distribution with rate parameter µ(R d )/(exp(M )⌫(R d )).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The CVP implements the-rich-get-richer dynamics as an interplay of these two preferential reinforcements, learning latent qualities of individual responses as inspired in Table 1.
Sentence 2: 3 max The {q T j } in the Table 3 shows the result from solving (1) up to T -th votes for each j 2 {1, 2}.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The 2N distances can determine the location of the vertices along the dimension up to a shift, through use of the law of cosines: For each subsequent dimension, two new vertices are chosen and new dissimilarities (δ ai , δ bi ) are computed by Dijkstra's algorithm.
Sentence 2: an MDS algorithm needs to be tested on distance matrices that are computed from distances between real points, in order to verify that the algorithm quickly produces sensible results.
Relationship: neutral
Domain: NeurIPS
Sentence 1: By Lemma 4, after t = 4(q−H(0)) r + 1 steps of the MW updates, H(t) > q, a contradiction.
Sentence 2: a mixed strategy of a Player i is a probability distribution over her strategy set, represented by a vector x i ∈ R ni with j∈Si x ij = 1, where x ij is the probability density that strategy j is chosen.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Moreover, µ is a minimizer of Eq.
Sentence 2: following ideas from [18], we can find a simple relaxation by considering the following constraints: (a) for all i 6 = j, for all i 6 = j, Y ij 6 inf{y i , y j }, which corresponds to x i x j 6 inf{x i , x j } for any x 2 [0, 1] n , (c) for all i, Y ii 6 y i , which corresponds to x 2 i 6 x i , and (d) for all (i, j) .
Relationship: neutral
Domain: NeurIPS
Sentence 1: When learning a function f , we may have access to not only the output values f (x i ) for training points x i , but also the values of its j-th order derivatives with respect to the input, D j x f (x i ).
Sentence 2: the stark difference in approximation error is highlighted in Figure 3, where we show the Styblinski-Tang function and its approximations with both regular and Sobolev Training.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The corpus has a high percentage of stops and fricatives in the initial and final consonant positions.
Sentence 2: the STI does not accurately model intraband masker non-linearities, phase distortions or the underlying auditory mechanisms (outside of independent frequency bands) We therefore sought to extend the AI/STI concepts to predict intelligibility, on the assumption that the closest physical variable we have to the perceptual variable of intelligibility is the auditory nerve response.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We consider the problem of approximate joint matrix triangularization when the matrices in M are jointly diagonalizable and real, but we only observe a set M' of noise perturbed versions of the matrices in M. Our main result is a first-order upper bound on the distance between any approximate joint triangularizer of the matrices in M' and any exact joint triangularizer of the matrices in M. The bound depends only on the observable matrices in M' and the noise level.
Sentence 2: the parameter σ can often be bounded by a function of the sample size, as in problems involving empirical moment matching [4].
Relationship: neutral
Domain: NeurIPS
Sentence 1: (taken from [16]) For a given sequence of Erdös-Rényi networks G(n, c n ), we have: As previously stated, our results hold for any given graph, and not only asymptotically.
Sentence 2: the first bound (Proposition 1) applies to any set of influencers A such that |A| = n 0 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: We analyzed the results in two ways: by counting each suffix once for each word type it was associated with, and by counting once for each word token (thus giving more weight to the results for frequent words).
Sentence 2: we will focus on a different preferential attachment scheme, based upon the two-parameter species sampling model [4,5] known as the Pitman-Yor process [6].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Indeed, iterative regularization algorithms are classical in inverse problems [15], and have been recently considered in machine learning [36,34,3,5,9,26], where they have been proved to achieve optimal learning bounds, matching those of variational regularization schemes such as Tikhonov [8,31].
Sentence 2: this excludes the choice t ⇤ (n) = 1 for all n 2 N ⇤ , namely considering only one pass over the data.
Relationship: neutral
Domain: NeurIPS
Sentence 1: On the WMT'14 English to French translation task, we obtained a BLEU score of 34.81 by directly extracting translations from an ensemble of 5 deep LSTMs (with 380M parameters each) using a simple left-to-right beam-search decoder.
Sentence 2: the right plot shows the LSTM's performance on sentences with progressively more rare words, where the x-axis corresponds to the test sentences sorted by their "average word frequency rank".
Relationship: neutral
Domain: NeurIPS
Sentence 1: We found the binomial approximation to yield the most accurate predictions in parameter ranges of interest to us, namely 500-4000 total units and connectivities ranging from .05-.15 (see Figure 2).
Sentence 2: in the presence of noise the same input applied multiple times to the same network will produce divergent trajectories , hence different final conditions, thus making the use of DALs for pattern classification problematic.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As mentioned earlier, most computation for the inside-outside algorithm is spent on the tensor calculation of T a→b c on the intermediate inside/outside quantities.
Sentence 2: finally, for vectors x, y, z ∈ R m , xy z is the tensor D ∈ R m×m×m where D i,j,k = x i y j z k (this is analogous to the outer product: [xy ] i,j = x i y j ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The rest of the components in the vector (98 altogether) were generated independently using the normal distribution with a zero mean and a standard deviation of 0.05.
Sentence 2: first note that given the vector v we can compute the norm of w as follows, Next, given two vectors x i and x j the value of their inner-product is, although we cannot compute the vector w explicitly we can still compute its norm and evaluate any of the kernels from the class C. Synthetic data: We generated binary-labelled data using as input space the vectors in 100 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The gap between VCL and batchnorm is larger for SELU and the lowest for ReLU, which is also consistent with the results in Tab.
Sentence 2: we are interested in distributions p ρ such that the quantity E[(σ 2 − σ 2 s ) 2 ] is minimized under the constraint that the variance is fixed, i.e., σ 2 = α.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the stereo-matching task subjects were presented with two target objects and a larger surrounding circle (25.2 degrees diameter) paired with one of the targets.
Sentence 2: the difference between uncertainty conditions was only significant for two surround depths (0.6 and 1.0 degrees, one-sided paired t-test p=0.004, p=0.0281) and not significant as a main effect (2-way ANOVA p=0.3419).
Relationship: neutral
Domain: NeurIPS
Sentence 1: On the other hand VI is subject to bias on account of the introduction of the approximating variational distribution.
Sentence 2: the following proposition shows that if each forward transition and backward transition are locally optimal, the result is globally consistent, which means the marginals r ⇤ t (z t ) and q ⇤ t (z t ) constructed by locally optimal transitions match the intermediate target f t (z t ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then, the new proof state constructed by unify is: Thus, the output score of the neural network S 0 ⇢ will be high if the subsymbolic representation of the input s is close to grandpaOf and the input i is close to BART.
Sentence 2: in addition, we constrain proofs to be cycle-free by checking whether a variable is already bound.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This perhaps suggests that it is only the exact mapping of the models to the neural substrate that holds any remaining theoretical interest.
Sentence 2: first the 'local field' is calculated as 3 + Input, then the output of the neuron is calculated as a threshold linear function of the local field: Thr and x i := 0 otherwise, where g := 0.53 a/ (1 − a) is the gain parameter, and h Thr := 0 is the threshold, and the value of k is set by iterative search to optimize performance.
Relationship: neutral
Domain: NeurIPS
Sentence 1: An overall very accurate approximation is then obtained if the subspaces are well selected and if they do contain the large majority of posterior mass.
Sentence 2: for a given observed data point, all except for a very small fraction of the latent components can be assumed to be non-causal or irrelevant, hence the corresponding latent space can be neglected for the integration over � s. For a sparse initiation (i.e., π � 1) of the spike-and-slab model ( 1) to (2), we consider such low dimensional subspaces to be spanned by a few (approximately πH) of the H latent space coordinates.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This association significantly broadens recent work connecting elementary, one-step iterative sparsity algorithms like (1) with simple recurrent or feedforward deep network architectures [20,30,34,38].
Sentence 2: one popular non-trivial instantiation of this approach assumes g(u) = i log (u i + ) with > 0 a user-defined parameter [10].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Therefore, during training we are masking the calculation of the unification success of a known ground atom that we want to prove.
Sentence 2: the constructed neural network allowed us to compute the gradient of proof successes with respect to vector representations of symbols, and thus enabled us to train subsymbolic representations end-toend from facts in a KB, and to induce function-free first-order logic rules using gradient descent.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The static strategy p t = 0 witnesses 0, T ∈ G T for every horizon T . To ensure R 1 T < T , any strategy will have to play p t > 0 at some time t ≤ T . But then it cannot maintain R 0 t = 0.
Sentence 2: suppose toward contradiction that R k t > r k at some t < T . An adversary may set all The two extreme trade-offs 0, T and T, 0 are Pareto optimal.
Relationship: neutral
Domain: NeurIPS
Sentence 1: More precisely, we wish to map from a (small) support set of k examples of input-label pairs S = {(x i , y i )} k i=1 to a classifier c S (x) which, given a test example x, defines a probability distribution over outputs ŷ.
Sentence 2: we compared our one-shot matching model to an oracle LSTM language model (LSTM-LM) [30] trained on all the words.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The R.EM 1110del of the \~lord-frequency effect described by Shiffrin and Steyvers [13,18,19] is a member of this class of models, as \ve describe next.
Sentence 2: hirshnlan et a1 found that the difference in the LF and H'F FA.Rs \¥as not affected b:y 1\1idazolam.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This can take Ω(n) time because regardless of where u is in the network, there always is another node v at distance Ω(n) from u.
Sentence 2: suppose that g decomposes into a product of simpler functions, g(X) = j f j (X j ), for some collection of real-valued functions f j , called factors, whose arguments are (index-ordered) sets X j ⊆ X.
Relationship: neutral
Domain: NeurIPS
Sentence 1: An early work that formally motivates various negative dependence conjectures is [41].
Sentence 2: a measure µ over 2 [n] is Exponentiated Strongly Rayleigh (ESR) if there exists an SR measure ν over 2 [n] and a power p ≥ 0 such that µ(S) ∝ ν(S) p . The parameter p serves to control the quality/diversity tradeoff by sharpening (p > 1) or smoothing out (p < 1) the variations of the ground SR measure (see Figure 1).
Relationship: neutral
Domain: NeurIPS
Sentence 1: If we did know |H 1 | (and equivalently know |H 0 | = n |H 1 |) then a FWER+FWPD algorithm is equivalent to the so-called top-k multi-armed bandit problem [9,8] and controlling FWER would be relatively simple using a Bonferroni correction:  Because we only know |S t | and not its expectation, the extra factors in the surrogate expression used in R t are used to ensure correctness with high-probability (see Lemma 7).
Sentence 2: [3,10]) we show that FDR is controlled for all times using the Benjamini-Hochberg procedure [11] (see Lemma 1), and also provide an anytime high probability bound on the false discovery proportion (see Lemma 2).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Most of these semi-supervised learning algorithms are applicable only to multiclass classification problems [1,10,32], with very few exceptions that develop discriminative models suitable for structured prediction [2,9,16,20,21,22].
Sentence 2: they proposed to minimize the following objective where the first term is the negative log conditional likelihood of the labeled data, and the third term is the conditional entropy of the CRF model on the unlabeled data.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This sPDE formulation has two chief advantages: it allows for a real-time synthesis using an AR recurrence and it allows one to recast the log-likelihood of the model as a generalization of the classical motion energy model, which in turn is crucial to allow for a Bayesian modeling of perceptual biases.
Sentence 2: observers sat 57 cm from the screen in a dark room.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Articles that are currently the best are shown in the serving bucket; others are explored in the random bucket until their scores are better than the current best articles; at this point they get promoted to the serving bucket.
Sentence 2: a data-driven machine learning approach can help with the scale issue, and we seek to blend the strengths of the editorial and algorithmic approaches by algorithmically optimizing content programming within high-level constraints set by editors.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our analysis in Section 6 of the full paper predicts that, once is small in comparison to the gap σ k σ k+1 − 1, we should see much more rapid convergence since q will depend on log(1/ ) instead of 1/ .
Sentence 2: as predicted, because its additional memory overhead and post-processing costs are small compared to the cost of the large matrix multiplication required for each iteration, Block Krylov Iteration outperforms Simultaneous Iteration for small .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let G be any compact group with identity e and ρ : G → C dρ×dρ be a unitary representation of G. Then given an array of possibly noisy and unsynchronized group elements, (g ji ) i,j and corresponding positive confidence weights (w ji ) i,j , the synchronization loss (assuming g ii = e for all i) To synchronize putative matchings between images, we instantiate this proposition with the approriate unitary representation of the symmetric group.
Sentence 2: the proposed model can easily generalize to other applications, outside computer vision, involving multi-matching problems.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Consequently, recent work has considered normalizing flows [28], in which random variables from (for example) a Gaussian distribution are fed through a series of nonlinear functions to increase the complexity and representational power of the encoder.
Sentence 2: considering Figure 2, note that VAE significantly underestimates the variance of the posterior (examining the marginals), a well-known problem of variational Bayesian analysis [7].
Relationship: neutral
Domain: NeurIPS
Sentence 1: This can be done using methods such as Ksubspaces [2], an extension of Kmeans to the case of subspaces, or Expectation Maximization for Mixtures of Probabilistic PCAs [3].
Sentence 2: the case in which n(t) = 1 and there are ρ 1 > 0, S ∈ N and i ∈ {1, . . . , N } such that for all m m+S t=m corresponds to the case of data that is rich in time.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The affinity between two nodes is computed as W ij,ij = exp( −|pi−p ′ j | 0.05 ), where p i is the orientation of normal vector at the sampled point (node) i to the contour, similarly to p ′ j .
Sentence 2: it can be obtained by setting the first derivative to zero ( ∂Φ(x,x) ∂x kl = 0), which gives we obtain the update rule in Eq.
Relationship: neutral
Domain: NeurIPS
Sentence 1: These predicates are useful for giving partial credit; for instance, it is easier to satisfy mod(y, 6) = 0 than y = 1296, but many programs that satisfy the former will have pieces that are also in the correct z.
Sentence 2: in the relaxed case, the Fisher information can be written to first order as I β , to first order, is the covariance of the penalty −β ψ with the second-order statistics of φ.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the experiments we discuss in the following, we set the hyperparameters as u The Bayesian method calculates p(θ, Z|X, M ), which is a posterior of unknown parameters and hidden variable set Z = (Y, S) given the data and model structure, M (in our case, this indicates the number of hidden states, and correlation structure).
Sentence 2: hMMs are suitable for such situations.
Relationship: neutral
Domain: NeurIPS
Sentence 1: All three cases (a)-(c), are polynomial optimization problems which we solve by first applying the sparse SDP relaxation of first order [19] with SDPA [20] as the SDP solver, and then applying IPOPT [21]  The results for the deterministic case (a) show that the optimal setting and the overall pressure sum i∈V1 w i differ even when the demand at only one node changes, as reported in Table 1.
Sentence 2: probability density functions can be readily estimated using a variety of data-driven methods, e.g., empirical histograms, kernel-based [8,9], and orthogonal basis [10] estimates.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The leading eigenvalue of its covariance matrix Σ is set as λ 1 = 100, and its corresponding eigenvector is sparse in the sense that only the first s = 5 entries are nonzero and set be to 1/ √ 5.
Sentence 2: for synthetic dataset II, we set s = 10 and k = 5.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We show how this improves the running time of gradient descent and SVRG.
Sentence 2: the worse-case time complexities of first-order methods are usually proportional to a fixed parameter times T . In large-scale settings, however, if we have already spent time computing the (full) gradient at x, perhaps we can use such information to reduce the time complexity to compute full gradients at other points near x.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In experiments on simulated dynamic systems, comparing our Gaussian approximation to Monte Carlo simulations, we found that the propagation method is comparable to Monte Carlo simulations, and that both approaches achieved more realistic error bars than a naive approach which ignores the uncertainty on current state.
Sentence 2: the iterated method provides any -step ahead forecast, up to the desired horizon, as well as the joint probability distribution of the predicted points.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We address this issue by introducing the notion of coset based multiresolution analysis (CMRA) on the symmetric group, find the corresponding wavelet functions, and describe a fast wavelet transform for sparse signals.
Sentence 2: this instance of L-CMRA is an exact analog of Haar wavelets: V k will consist of all functions that are constant on each left S n−k -coset.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We describe the basic ideas of LP in Section 2, show how it applies to Gaussian Processes (in particular the Bayes Committee Machine of [9]) in Section 3, prove that SVM chunking is a special case of LP in Section 4, and finally demonstrate in experiments the feasibility of LP (Section 5).
Sentence 2: we thank Nir Friedman, Zoubin Ghahramani and Adam Kowalczyk for useful suggestions and discussions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The consistency problem has been studied before as the membership problem for the marginal polytope (see Related Work); in the case of inconsistency, the closest consistency problem seeks the minimal perturbation to the data necessary to recover coherence.
Sentence 2: by strong duality, the corresponding "reduced primal linear program," obtained from the linear program in Lemma 3.2 by retaining only the decision variables corresponding to the K 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then, Algorithm 1 (where OPTCARD is replaced with Approximate OPTCARD, Algorithm 2) runs in at most log(k) + log(Q)/λ + 1 rounds and produces a solution of size at most 2k + 2 log(Q)k/λ .
Sentence 2: a gd [m, ] = DISCARD(V, m, , A).
Relationship: neutral
Domain: NeurIPS
Sentence 1: To do so, we define the regularized empirical risk R n (w) := L n (w) + (cV n /2)kwk 2 and the corresponding optimal argument and attempt to minimize R n with accuracy V n . Since the regularization in ( 4) is of order V n and (3) holds, the difference between R n (w ⇤ n ) and L(w ⇤ ) is also of order V n -this is not immediate as it seems; see [16].
Sentence 2: this bound not only improves the bound O(N 1+↵/2 ) for Ada AGD, but also enhances the complexity of SVRG for reaching the same target accuracy which is given by O In this section, we compare the adaptive sample size versions of a group of first-order methods, including gradient descent (GD), accelerated gradient descent (AGD), and stochastic variance reduced gradient (SVRG) with their standard (fixed sample size) versions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: These results thus fall in line with prior work demonstrating consistency of resting state networks across subjects [35,12], and reveal that feature-driven activity during natural viewing induces a similar functional clustering as that occurring during rest.
Sentence 2: figure 1 shows the weights for the motion variable, Figure 2 for the human body variable, and Figure 3 for the language variable.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The strength of such a fully Bayesian approach lies largely in its extensibility to allow for more accurate signal and noise models; in this vein work is continuing on the development of appropriate conditional prior structures for audio signals, including the formulation of Markov random field models.
Sentence 2: consider a function ¡ whose time-frequency support is centred about the origin, and let ¡ ¡ ¢ £ denote a time-shifted (translation by ¤ ¦¥ ) and frequency-shifted (modulation by § ¨) version thereof; such a collection of shifts defines a sampling grid over the time-frequency plane.
Relationship: neutral