sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The proposed probabilistic population coding (PPC) framework provides a statistically efficient neural representation of probability distributions that is both broadly consistent with physiological measurements and capable of implementing some of the basic operations of probabilistic inference in a biologically plausible way.
Sentence 2: these experiments and the corresponding neural models have largely focused on simple (tractable) probabilistic computations such as cue combination, coordinate transformations, and decision making.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The inversion can therefore also be expanded according to the Neumann series; i.e., θ = ∞ i=0 [γλ] i r. Similar to other kernel-based policy evaluation methods [23,24], NPDP has a computational complexity of O(n 3 ) when performed naively.
Sentence 2: by taking advantage of sparse matrix computations, this complexity can be reduced to O(nz), where z is the number of non-zero elements in (I − γλ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, gradient computation for L 2 smoothing requires sorting factors rather than just maximizing over their values, incurring worst-case cost of O(W r log W r ) per factor instead of just O(W r ) for soft-max gradient.
Sentence 2: one can use partitioning around a pivot value instead of sorting, yielding O(W r ) cost in expectation [3], and caching the pivot can also speed-up the runtime considerably.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A naive simple Monte Carlo estimator of the KL divergence between the output distributions of two inference algorithms requires evaluating output probabilities for both algorithms.
Sentence 2: it is typically intractable to compute output probabilities for sampling-based inference algorithms like MCMC and SMC, because that would require marginalizing over all possible values that the random variables drawn during the algorithm could possibly take.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The rationale is that more rapid sampling can reduce the variance of a Monte Carlo estimate and hence outweigh the bias introduced.
Sentence 2: the added flexibility introduces new challenges for sampler and parameter selection, since standard sample quality measures, like effective sample size, asymptotic variance, trace and mean plots, and pooled and within-chain variance diagnostics, presume eventual convergence to the target [1] and hence do not account for asymptotic bias.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For setting (i), where the data lies in R D , the estimation rate has a dependence on the dimension D, which may be unattractive in high-dimensional situations: this is known as the curse of dimensionality, suffered by density-based procedures in general.
Sentence 2: when the data has low intrinsic dimension in (ii), it turns out that, remarkably, without any changes to the procedure, the estimation rate depends on the lower dimension d and is independent of the ambient dimension D. in realistic situations, the data may not lie exactly on a lower-dimensional manifold, but near one.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Here we have only studied variants of CAL (and its noise-robust generalization).
Sentence 2: one could also apply this style of analysis to other methods, to investigate analogous questions of how the label complexities of such methods degrade as the batch sizes increase, or how such methods might be modified to account for a sublinear cost function, and what results one might obtain on the total cost of learning with these modified methods.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For any positive ε > 0, there exists a positive numerical constant C such that if entries are revealed with probability , where σ max max σ and σ min min σ , then the following holds with probability at least 1 − n −5 log 2 (4 √ r T F /ε): • the problem (3) has a unique optimal solution; and .
Sentence 2: for simplicity of notations and to emphasize key points of our proof, we only focus on 3-mode tensors in Section 2.3.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: On the other hand, off-policy reinforcement learning (RL) concerns the problem of learning a good (or possibly better) policy with data collected from a behavior policy [Sutton and Barto, 1998].
Sentence 2: to the best of our knowledge, previous methods do not have satisfiable performance or are not directly applicable in a complex environment as ours with continuous state and hybrid action space.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Let δ t be the step size at iteration t. The algorithm performs the following updates: The above updates require computation of K 1/2 which is expensive for large high-rank matrices.
Sentence 2: using elementary linear algebra we can show that K and the learned kernel function can be computed efficiently without computing K 1/2 by maintaining S = K −1/2 KK −1/2 from step to step.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Our problem then becomes: Since the log likelihood scales linearly with the size of the data, and the singular values scale with the square root of the size, we also add a factor of √ nT in front of the nuclear norm term.
Sentence 2: in the examples in this paper, we assume spikes are drawn from a Poisson distribution: this method can be used with any point process with a log-concave likelihood.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In this study, the steganalysis networks were trained and used after the complete encoding system was created.
Sentence 2: using pre-trained and/or simultaneously trained steganalysis networks in an adversarial learning framework, such as Generative-Adversarial-Networks (GAN) [30], during the training of steganography nets provides a method to incorporate an obfuscation based error metric.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Language comprehension in humans is significantly constrained by memory, yet rapid, highly incremental, and capable of utilizing a wide range of contextual information to resolve ambiguity and form expectations about future input.
Sentence 2: most of the leading psycholinguistic models and fielded algorithms for natural language parsing are non-incremental, have run time superlinear in input length, and/or enforce structural locality constraints on probabilistic dependencies between events.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, such analysis will rely on a bound on the variance of the local gradients (as apposed to the bound in equation ( 2)), which is a strong assumption that ignores the effect of averaging over the P nodes.
Sentence 2: assumption 1 allows for a more elegant analysis that provides better convergence rates, which are due to the averaging of the local gradients at every step of the TopK algorithm.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The probability distribution over the elements y nj is defined as previously in the limit construction of the IBP: ).
Sentence 2: the lower binary variables z nk are also defined as Bernoulli distributed random quantities: here the probability that z nk = 1 is a function of the upper binary variables y n,: and the kth column of the weight matrix V , with probabilities V jk ∈ [0, 1] connecting y nj to z nk .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The integrality of the bipartite LP relaxation is not guaranteed, however it can be enforced by adding the so-called blossom inequalities [22]: where L ⊂ 2 V is a collection of odd cycles in G, called blossoms, and δ(S) is a set of edges between S and V \ S. It is known that if L is the collection of all the odd cycles in G, then LP (2) always has an integral solution.
Sentence 2: notice that the number of odd cycles is exponential in |V |, thus solving LP ( 2) is computationally intractable.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Even slight random shifts in the pixel-derived representation of less than 0.25 • of visual angle (on the order of micro-saccades) dramatically reduced the fits to 0.25 (G) and 0.21 (S).
Sentence 2: the same random shifts did not change the average fit numbers for the combination of dorsal and ventral models (0.53 for G, 0.39 for S).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For a matrix M, we define a risk function R : When X has covariance matrix Σ, R(w; Σ) = Var(w T X) is the variance of the portfolio return, w T X, and is employed as the objected function in the GMV formulation.
Sentence 2: estimating Σ is difficult due to the heavy tails of asset returns.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: While the exact solution in the Gaussian case can be computed in polynomial time, estimating an Ising model via maximum likelihood in general is N P-hard .
Sentence 2: as a consequence of applying the screening rule, the blockwise structure of an 1 -regularized Ising model can be determined as easily as the blockwise structure of a Gaussian graphical model, despite the fact that within each block, exact learning of a sparse Ising model could still be challenging.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Let O = S 1 ∩ S 2 be the indices for which both sensors are available, and By using the samples in S 1 and S 2 as two independent training sets, we may obtain two separate classifiers (denoted w 1 and w 2 ).
Sentence 2: we can coordinate the information from both sensors by using an idea known as co-training [4]: on the O U samples, classifiers w 1 and w 2 should agree as much as possible.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: T π is defined as Taking again a pessimistic approach, the one-step Bellman-error of F is defined as Typically by increasing F, E 1 (F, Π) can be made smaller (this is discussed at some length in [3]).
Sentence 2: it also holds for both Π and F that making them bigger will increase their capacity (pseudo-dimensions) which leads to an increase of the estimation errors.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In our case, interconnected brain regions can be considered as nodes of a probabilistic graphical model and interactivity relationships between regions are modeled by probability values on the arcs (or the lack of) between these nodes.
Sentence 2: the major challenge in such a machine learning approach is the choice of a particular structure that models connectivity and interactivity between brain regions in an accurate and efficient manner.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: An alternative solution is to use the Shannon entropy of each column of the matrix factor as sparsity regularizer [7], since a vector with unit 1 norm and low entropy implies that only a few of its components are significant.
Sentence 2: the entropic prior based regularizer does not afford a closed form solution, and an iterative fixed point algorithm is described based on the Lamber's W-function in [7].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Second, since the data X(k) might not satisfy the linear ICA assumption, the optimal linear transform might not transform X(k) into independent coordinates.
Sentence 2: it does transform X (k) into the least dependent coordinates, since Further more, if the linear transform A is constrained to be orthogonal, then finding the least dependent coordinates is equivalent to finding the marginally most non-Gaussian coordinates, since (notice that the negentropy is invariant under orthogonal transforms).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, with a single epoch of training H-RLS does not perform worse than SVM (except on OHSUMED under the normalized H-loss) and comes reasonably close to H-SVM.
Sentence 2: the performance of APPROX-H-BAYES is disappointing: on OHSUMED it is the best algorithm only for the uniform H-loss, though it was trained using the normalized H-loss; on RCV1 it never outperforms H-RLS, though it always does better than PERC and H-PERC.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Furthermore, the construction of kernel induced feature spaces, lends itself to efficient implementation of distance measures over spike trains that are better suited to comparing two neural population trajectories than the Euclidean distance in the original space of spike counts per bins [11,5].
Sentence 2: sVR is a "static" method that does not take into account the smooth dynamics of the predicted movement trajectory which imposes a statistical dependency between consecutive examples.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As a simple example in the binary classification setting, if p(x) is supported on two disjoint sets and if P (Y = 1|X = x) is strictly greater than 1/2 on one set and strictly less than 1/2 on the other, then perfect knowledge of the decision sets reduces the problem to a hypothesis testing problem for which ǫ 2 (n) = O(e −ζ n ), for some constant ζ > 0.
Sentence 2: if γ is small relative to the average spacing n −1/d between labeled data points, then ǫ 1 (n) = cn −1/d where c > 0 is a constant.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Our model, along with various others, addresses these issues at a cortical level via two basic mechanisms: DA modulation, which changes the excitability of neurons in a particular way (units that are inactive are less excitable by input, while units that are active can become more active), and targeted input from the BG.
Sentence 2: models differ as to the nature and provenance of the BG input, and also its effects on the PFC.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: LRR is an important robust subspace clustering technique and has found wide applications in machine learning and computer vision, e.g., motion segmentation, face clustering, and temporal segmentation [12,14,6].
Sentence 2: the existing LRR solver [12] is based on ADM, which suffers from O(n 3 ) computation complexity due to the matrix-matrix multiplications and matrix inversions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Again, we observed (Figure 3-top left) that the JS-divergence estimate remains close to 1 while training DCGAN with BN.
Sentence 2: after applying two different Lipschitz regularization methods, SN and GP in Figures 3-top right and bottom left, we observed that the hybrid d JSD,W1 changed nicely and monotonically, and correlated well with the quality of samples generated.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For any sequence of querying rules and any autonomous user indexed by {u j } j2[M ] , there is a fixed constant c 0 such that the cost to the user satisfies Proof.
Sentence 2: ideally, none of the M queries causes a premature halt, and the total cost is at most 96 ⌧ 2 (1 + ln(M )), but the adaptive user may cause rounds to end prematurely and pay up to 6N t . by Lemma 3, with probability 1 t if one of the adaptive user's queries causes a round t to end prematurely, then the amount of data, N t , and the number of the user's queries answered in that round, M t , must satisfy Given M , there is a largest t for which this is possible since N t = 3 t N 0 and t = 2 t 0 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: With enhanced flexibilities, these recent approaches are often good at assessing individual risks, such as predicting a patient's hazard function or survival time.
Sentence 2: except for the Gaussian process whose results are not too difficult to interpret for low-dimensional covariates, they often have difficulty in explaining how the survival is impacted by which covariates, limiting their use in survive analysis where interpretability plays a critical role.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Among the most commonly used priors are Gaussian process (GP) [24], Dirichlet process (DP) [2] and Indian buffet process (IBP) [12].
Sentence 2: standard nonparametric Bayesian models are limited in that they usually make very strict and unrealistic assumptions on data, such as that observations being homogeneous or exchangeable.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The doubling trick provides one way to adapt the learning rate; indeed, we observe that the regret of Hedge with the doubling trick is initially smaller than the regret of Hedge with fixed learning rate.
Sentence 2: unlike AdaHedge, the algorithm never detects that its current value of η is working well; instead it keeps exhausting its budget, which leads to a sequence of clearly visible bumps in its regret.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The points are assigned to clusters according to a probability distribution indicating the relative sizes of the clusters.
Sentence 2: not only in our model f + and f − can be arbitrary probability mass functions (pmfs), we do not have to make any assumption on k or the cluster size distribution, and can allow for any partitioning of the set of elements (i.e., adversarial setting).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When the spectrum of M is light-tailed (so that ρ T d and, simultaneously, det κ (M ) 1), with the smaller eigenvalues λ i corresponding to eigenvectors in which f is smoother (so that the ratios ∇ ui f 2 ∞ λ i remain controlled), then our bound improves on (2).
Sentence 2: when no preliminary knowledge about good f is available, we may run the algorithm with M equal to the identity matrix and recover exactly the bound (2).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Writing Ĉt−1 = With such covariance matrices, computing the (regularized) natural direction v t is equal to (11) Using the Woodbury identity with positive definite matrices [7], we have is of size p × r (with r < p, thus yielding a covariance matrix of rank r), the cost of this computation is O(pr 2 + r 3 ).
Sentence 2: since the Gram matrix G t = X T t X t can be rewritten as the cost of computing G t using G t−1 reduces to O(pr + r 3 ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It illustrates that the saliency values of unseen natural image patches usually lie between −2.0 and 1.8 (for the training data, they are between −1.8 and 2.2).
Sentence 2: our optimal stimuli have saliency values of 5.0 and 5.5, indicating that they represent the difference between fixated and background locations in a much more articulated way than any of the noisy measurements in our data set.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We fix d = 3 throughout to avoid having too many parameters to vary, although similar results are obtained for larger d. Exact prediction of learning curves by analytical calculation is very difficult due to the complicated way in which the random selection of training inputs enters the matrix K and vector k in (9).
Sentence 2: by first expressing these quantities in terms of kernel eigenvalues (see below) and then approximating the average over datasets, one can derive the approximation [3, 6] This equation for has to be solved self-consistently because also appears on the r.h.s.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If in addition to r1 also q1 is known, the two local minimia of F ∞ (p 1 = 0 and p2 = 0) allow to recover t0 = t 0 only.
Sentence 2: if p1 = p 1 (or p2 = p 2 ) is known exactly, there are three local minima again-p 2 = 0, q1 = 0, r1 = 0-but now none of effective parameters is equal to its true value: ti = t i (i = 0, 1, 2).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As for the transition function, although it is clear that the brain contains detailed representations of action-outcome associations [19], their anatomical localization is not yet entirely clear.
Sentence 2: some evidence suggests that the enviromental effects of simple actions may be represented in inferior fronto-parietal cortex [20], and there is also evidence suggesting that medial temporal structures may be important in forecasting action outcomes [21].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For groups not containing the neuron with the largest input, this condition can be satisfied if and only if they are in the top c groups .• The winner-take-all competition described above holds only for the case of strong inhibition (3 > 1 -a.
Sentence 2: if (3 is small, the competition will be weak and may not result in group-winner-take-all.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We may assume that µ is a sub-Gaussian probability measure on R D (see §2.1 and the extended version of this paper).
Sentence 2: in order to easily express the dependence of our probabilistic estimates on properties of the measure µ, we assume for simplicity that µ is compactly supported and denote by R µ the minimal radius among all balls containing the support of µ, that is, where B(0, r) is the ball around the origin 0 with radius r. We further assume that for some 0 < γ < 1, µ satisfies the following condition, which we refer to as the "two-subspaces criterion" (for γ): For any pair of (D − 1)-dimensional subspaces of R D , L 1 and L 2 : We note that if µ satisfies the two-subspaces criterion for any particular 0 < γ < 1, then its support cannot be a union of two hyperplanes of R D . The use of this assumption is clarified below in §3.2, though it is possible that one may weaken it.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The augmented model raises the cross-validation score of model (c) to 1 bit/s, meaning that by incorporating dependencies between intervals, the model carries slightly more predictive information than a homogeneous Poisson model, despite the mis-specified stimulus filter.
Sentence 2: this model-despite passing time-rescaling tests of both marginal distribution and independence-still carries less information about spike times than the inhomogeneous Poisson model (b).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Prior work on classifier adaptation [23,24,25] assumes that f t (x) = f s (x) + ∆f (x), where the perturbation ∆f (x) is a function of input feature x.
Sentence 2: these methods require target labeled data to learn the perturbation function, which cannot be applied to unsupervised domain adaptation where target domain has no labeled data.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, one can think of losses as negative log-likelihoods in a conditional model that generates each successive layer given its predecessor; i.e., L 1 (W x, ) = log p W ( |x) and L 2 (V , y) = log p V (y| ).
Sentence 2: (a loss based formulation is more flexible, since every negative log-likelihood is a loss but not vice versa.)
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Setting aside architectural specifics, the vast majority of VQA models operate on a set of similar design principles -first producing vector representations for the image and question and then combining them to predict the answer (often through complex attention mechanisms).
Sentence 2: when language biases are quite strong, the question feature may already be sufficiently discriminative and the model can learn to ignore the visual signal without facing significant losses during training (e.g.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Several Poisson-based MF models were proposed recently to model large and sparse static ratings, e.g., Poisson Factorization (PF) [25], and collaborative topic PF for modeling content-based recommendations [27]; and dynamic data, e.g., dynamic PF (dPF) [16], dynamic compound PF (DCPF) [34], and deep dynamic PF [24].
Sentence 2: none of these can effectively and efficiently handle massive, dynamic and sparse ratings simultaneously (see more analysis in Section 2).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Under Assumptions 1 and 2, the iterates of CDSGD satisfy ∀k ∈ N: At a high level, since E[∇J i (x k )] is the unbiased estimate of ∇V (x k ), using the updates ∇J i (x k ) will lead to sufficient decrease in the Lyapunov function.
Sentence 2: unbiasedness is not enough, and we also need to control higher order moments of ∇J i (x k ) to ensure convergence.
Relationship: contrasting