sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: In the binary classification setup, learning with complementary labels is equivalent to learning with ordinary labels, because complementary label 1 (i.e., not class 1) immediately means ordinary label 2.
Sentence 2: in K-class classification for K > 2, complementary labels are less informative than ordinary labels because complementary label 1 only means either of the ordinary labels 2, 3, . . . , K. The complementary classification problem may be solved by the method of learning from partial labels [5], where multiple candidate class labels are provided to each training pattern-complementary label y can be regarded as an extreme case of partial labels given to all K 1 classes other than class y.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This equivalent formulation suggests a naive sampling (NS) algorithm for approximating σ(A, T ): draw n samples of {τ ji } (j,i)∈E , run a shortest path algorithm for each sample, and finally average the results (see Appendix C for more details).
Sentence 2: this naive sampling approach has a computational complexity of O(nC|V||E| + nC|V| 2 log |V|) due to the repeated calling of the shortest path algorithm.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This coefficient can be arbitrarily large: consider the case where µ concentrates on a single starting state (that is µ(s 0 ) = 1 for a given state s 0 ) and such that the optimal policy leads to other states (that is, d µ,π * (s 0 ) < 1), the coefficient is then infinite.
Sentence 2: it is also the best concentrability coefficient according to [21], that provides a theoretical and empirical comparison of Approximate Policy Iteration (API) schemes.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Implications for Portfolio Selection: By comparing Propositions 1 and 2, the first interesting conclusion one can reach about portfolio selection is that, without considering any additional information, the worst-case CVaR criterion yields the same optimal portfolio weighting x as the worst-case VaR criterion (recall that VaR minimizes α in Proposition 1 by adjusting x while CVaR minimizes f by adjusting x in Proposition 2).
Sentence 2: the worst-case distributions for the two approaches are not the same, which can be seen from the relation ( 16) between VaR and CVaR and observing that α in ( 4) is not the same as in (19).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then our theorem indicate that as long as m = poly(N, 1/δ ), where δ is the minimal distance between each examples, we can actually fit arbitrary labels of the input data.
Sentence 2: since the total iteration only depends on log m, when m = poly(N, 1/δ ) but the input data is actually structured (with small k, l and large δ), then SGD can actually achieve a small generalization error, even when the network has enough capacity to fit arbitrary labels of the training examples (and can also be done by SGD).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In the traditional BN, the normalization step seeks to eliminate the change in the distributions of its internal layers, by reducing their internal covariant shifts.
Sentence 2: prior to normalizing the distribution of a layer, BN first estimates its BN has a limitation, which is limited by the memory capacity of computing platforms (e.g.GPUs), especially when the network size and image size are large.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Given diverse object appearances and correlations in each image, personalized edge connections with respect to each knowledge form should be adaptive for different regions.
Sentence 2: our work aims to develop in-place knowledge modules which can not only explicitly incorporate any kinds of commonsense knowledge (both explicit or implicit) for better semantic reasoning but also link external knowledge with visual observations in each image in an adaptive way.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: RRL has not yet demonstrated scalability to problems as complex as those considered here-previous RRL blocks-world experiments include relatively simple goals 13 , which lead to cost functions that are much less complex than the ones here.
Sentence 2: unlike RRL, our API assumes an unconstrained simulator and (for the FF heuristic) a world model, which must be provided or learned by additional techniques.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Recently a generic acceleration algorithm called Catalyst [Lin et al., 2015], based on the proximal point method improved this rate of convergence, but the practical performances highly depends on the input parameters.
Sentence 2: recent papers, for example [Shalev-Shwartz and Zhang, 2014] (Accelerated SDCA) and 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In fact, a natural greedy algorithm gives a factor 1/2 approximation for min-min, while local search gives a constant factor approximation for sum-sum, even with matroid constraints [6,2,4].
Sentence 2: for the sum-min objective, the best known algorithm had an approximation factor of O(1/ log n) [6] and no inapproximability results were known.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Now construct h 2 , also with sparsity r 2 = 5 + 5/ε.
Sentence 2: the loss for H our batch algorithm uses sparsity r = 10 + 10/ε in each encoder h 1 , h 2 and achieves reconstruction error (1 + ε) X − X 2 2 F . The iterative algorithm uses sparser features, but pays for it a little in reconstruction error.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: More importantly, one can encode structural information through the cost function F , which selects and establishes preferences over the set of atoms S F . As pointed out in [6], when F is submodular, (4) can be evaluated by a secant method with submodular minimizations ([21, §8.4], see also Appendix B).
Sentence 2: as we will show, it is possible to do significantly better by completely avoiding submodular optimization.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In using all of the eigenvectors, the dimensionality of the feature embedding is not reduced.
Sentence 2: in scaling the eigenvectors by the inverse square-roots of their eigenvalues, the directions of least cost in the network regularized problem become the directions of maximum variance in the associated ridge regularized problem, and hence are the directions of least cost in the ridge problem.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We would ideally like to represent a given observation x by the unknown θ (or the corresponding a related to θ by θ = aV + b) that generated it, since the conditional distribution P (x|θ) is used to model random effects.
Sentence 2: the actual value of a is not known to us and all of our knowledge of a is contained in the posterior distribution P (a|x) = P (a)P (x|a)/P (x).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, these two approaches differ in an important point: Sherali-Adams hierarchy uses beliefs over all subsets of |R|  d variables, where d is the degree in the hierarchy; this leads to an LP of size ).
Sentence 2: gBP only retains regions that are contiguous in G. If G has maximum degree k, this produces an LP of size O(nk d ), a reduction which is significant for large-scale problems.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The best known such constant is due to the work of [17] that requires a bound on the RIP constant δ 2k ≤ 0.5 (or equivalently a bound 1+δ 2k 1−δ 2k ≤ 3 on the condition number).
Sentence 2: real-life high dimensional statistical settings, wherein pairs of variables can be arbitrarily correlated, routinely require estimation methods to perform under arbitrarily large condition numbers.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: From Proposition 4, we know that the two inequalities are strict.
Sentence 2: if φ is top-2 subgroup order sensitive, the following inequalities hold with at least one of them being strict: φ(g, (1, 2, 3)) ≥ φ(g, (2, 1, 3)), φ(g, (1, 3, 2)) ≥ φ(g, (2, 3, 1)), and φ(g, (3, 1, 2)) ≥ φ(g, (3, 2, 1)).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We then use the plugin estimator How well does this plugin estimator work, that is, what is its relative risk We cannot simply write L n ( λn ) and apply Theorem 1 because L(•) can only be applied to nonrandom arguments.
Sentence 2: we can still leverage existing machinery by defining a new plugin regularizer R Henceforth, the superscript • will denote quantities concerning the plugin regularizer.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Unfortunately, it is more challenging to give a simple estimate of the optimal k as in Theorem 3.4, even when A t is a random bipartite graph, because the spectral properties of the random matrix are complicated by blockwise structure, and may depend on the prior distribution Q(ν).
Sentence 2: experimentally the optimal k follows the trend a n t , where the constant a depends on both the model assumption and the choice of Q(ν), and can be numerically estimated by simulation.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: More precisely, we consider prior distribution P with probability density P (z i , σ) to be facotorizable in its compression set dependent component and message string component (conditioned on a given compression set) such that: The above choice of the form for P I (i) is appropriate since we do not have any a priori information to distinguish one compression set from other.
Sentence 2: as we will see later, we should choose p(d) such that we give more weight to smaller compression sets.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: One of the key advantages of a standard Bloom filter is that it is easy to insert an element (possibly slightly changing the false positive probability), although one cannot delete an element without using a more complex structure, such as a counting Bloom filter.
Sentence 2: there are more recent alternatives to the standard Bloom filter, such as the cuckoo filter [6], which can achieve the same or better space performance as a standard Bloom filter while allowing insertions and deletions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note that the schizophrenia patients studied here have been selected for their prominent, persistent, and pharmaco-resistant auditory hallucinations [14], which might have increased their clinical homogeneity.
Sentence 2: the patient group is not representative of the full spectrum of the disease, and thus our conclusions may not necessarily apply to all schizophrenia patients, due to the clinical characteristics and size of the studied samples.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: One common solution to this problem is to perform validation on fewer number of samples, where the downside is that the learner would obtain a much more noisy and sometimes completely unstable estimate of the out-of-sample performance compared to the case where the entire LOOCV vector is at the learner's disposal.
Sentence 2: aLOOCV described next will provide the benefits of LOOCV with negligible additional computational cost.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The main challenge is to evaluate p(✓ d |w d,1:N , , ↵), i.e., infer the topic proportion given each document, which is also the important inference problem in the unsupervised LDA model.
Sentence 2: it is well known that the exact evaluation of the posterior probability p(✓ d |w d,1:N , , ↵) is intractable [4,5,9,15,26,27].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Such linear transformations seem to suggest linear dendritic operations and, in combination with a single global 'somatic' nonlinearity, they are often assumed in neural network models and descriptive models of neuronal signal processing [10].
Sentence 2: as we will show below, estimation from the spike trains of multiple correlated presynaptic neurons requires a non-linear integration of inputs even in this case.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The first part of the regret (i.e., the regret coming from pulling the suboptimal arms) suggests that the exploration phase τ should be long enough for the algorithm to select the empirically best arm î * at τ equivalent to the actual optimal arm i * with high probability; and at the same time, as short as possible to reduce the number of times the suboptimal arms are explored.
Sentence 2: the second part of the regret (i.e., the variance of pulling arms with different means) is minimized by taking τ as small as possible (e.g., τ = 0 would guarantee a zero regret).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For small amounts of poisoned data ( 5%) we can still certify the resilience of MNIST-1-7 and Dogfish (Section 5).
Sentence 2: with more (30%) poisoned data, the attacker can subvert the outlier removal to obtain stronger attacks, increasing test error on MNIST-1-7 to 40%-much higher than the upper bound of 7% for the oracle defense.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Since the generalization error is not known, it is tempting to try to estimate the optimal convex aggregate classifier by minimizing the training error P n {(x, y) : yf (x) ≤ 0} over the convex hull conv(H).
Sentence 2: this minimization problem is not computationally feasible and, moreover, the accuracy of empirical approximation (approximation of P by P n ) over the class of sets {{(x, y) : yf (x) ≤ 0} : f ∈ conv(H)} is not good enough when H is a large class.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Even worse, Baik, Ben-Arous and Péché [BBAP05] and Paul [Pau07] demonstrate a phase transition phenomenon: if β < √ α the estimate is asymptotically orthogonal to the signal v PC , v → 0.
Sentence 2: for β > √ α, v PC , v remains strictly positive as n, p → ∞.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: On unweighted Let C H(i t ) be the set of the connection nodes i of H(i t ) for which ∆(i) = 0 if C H(i t ) ≡ ∅ Let j be the node of C H(i t ) closest to i t Set y it = sgn ∆(j) else Set y it = −1 (default value) trees, computing ∆(i) for a connection node i reduces to the Fork Label Estimation Procedure in [5, Lemma 13].
Sentence 2: predicting with the label of the connection node closest to i t in resistance distance is reminiscent of the nearest-neighbor prediction of WTA on weighted line graphs [6].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In the discriminative optimization problem formulated in Section 4.1, the µ variables are used to optimistically select both instances and their labels, with the goal of achieving the best classification model according to the objective (5).
Sentence 2: when the labeled set is small and the discriminative partition (clustering) information contained in the large unlabeled set is inconsistent with the true classification, the labels optimistically guessed for the selected instances through µ might not match the underlying true labels.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: [5] give another bound on window size which requires τ = O(n/m).
Sentence 2: with a output alphabet of size m, specifying all moments in a N length continuous time interval requires m N time and samples, and therefore all of these approaches lead to exponential runtimes when m is constant with respect to n. Also relevant is the work by Anandkumar et al.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The most famous approach is spectral clustering [7], which corresponds to the spectral relaxation of the ratio/normalized cut and uses k-means in the embedding of the vertices found by the first k eigenvectors of the graph Laplacian in order to obtain the clustering.
Sentence 2: the spectral relaxation has been shown to be loose for k = 2 [8] and for k > 2 no guarantees are known of the quality of the obtained k-cut with respect to the optimal one.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: factor of 1 κg (1 − ( In the worst case, k f = 1 and K f = n, in which case the guarantee is 1/n.
Sentence 2: the bound above follows from a simple observation that the constraint {f (X) ≤ b} is down-monotone for a monotone function f . in this variant, we do not use any specific information about f . In particular it holds for maximizing a submodular function g over any down monotone constraint [2].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The resulting complete-information game has i∈N |Θ i | players and |A i,θi | actions for each player (i, θ i ); a Nash equilibrium can be represented using just i θi |A i,θi | numbers.
Sentence 2: the normal form representation of the agent form has size j∈N |Θ j | i,θi |A i,θi |, which grows exponentially in n and |Θ i |.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, this requires us to evaluate the intractable normalization constant during training as well as in the inference, making the model not tractable anymore.
Sentence 2: we may consider using the log-probability of a sample under the mixture-ofexperts model as the training criterion (18) This criterion resembles clustering, where individual models may specialize in only a fraction of the data.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We are operating in a non-Bayesian setting, so we are not given a measure (prior) on the space H. In the absence of a measure, there is no natural notion of the "volume" of the current version space.
Sentence 2: the distribution P does induce a natural distance function on H, a pseudometric: We can likewise define the notion of neighborhood: We will be dealing with a separable learning scenario, in which all labels correspond perfectly to some concept h * ∈ H, and the goal is to find h ∈ H such that d(h * , h) ≤ ǫ.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Suppose first that neither endpoint of P is a leaf of T e (t).
Sentence 2: then, from parts 1 and 3 of Lemma 1 it follows that from part 2 of Lemma 1 it follows that Now by construction the root e ∈ P ∩ S, and hence w(P ∩ M * T ) > w(P ∩ M ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In image analysis, the transformed mixture of Gaussians (TMG) model uses a probabilistic framework and an EM algorithm to jointly learn clustering and alignment of image patches subject to various forms of linear transformations [4].
Sentence 2: this model only considers sets of transformations in discrete pixel space, whereas we are focused on curve modelling that allows for arbitrary continuous alignment in time and space.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The above sequence of inequalities give us that inf Since this holds for all p ∈ Δ n , we have that We note that Ramaswamy and Agarwal [16] showed a similar least-squares type surrogate calibrated for any loss � : Y × T →R + ; indeed our proof technique above draws inspiration from the proof technique there.
Sentence 2: the surrogate they gave was defined on a surrogate space of dimension n − 1, where n is the number of class labels in Y.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As a consequence, the recovery algorithm for p and M is identical to theirs.
Sentence 2: insofar as the asymmetric part of the kernel is concerned, everything, starting from the definition and the introduction of the vector field r as a way to model the asymmetry, through the derivation of the asymptotic expression for the symmetric plus asymmetric kernel, is new.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Optimization (2) shares the same underlying insight with Self-Paced Learning (SPL) [14,15] where the algorithm automatically learns the weights over data points during training.
Sentence 2: the process and scope in the two methods differ fundamentally: SPL minimizes the weighted loss over datapoints within a single domain, while optimization (2) minimizes the weighted loss over multiple tasks across possibly heterogeneous domains.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The benefits of the GAMP method for neural mapping are that it is computationally tractable with large sums of data, can incorporate very general graphical model descriptions of the neuron and provides a method for simultaneously estimating the parameters in the linear and nonlinear stages.
Sentence 2: methods such as the common spike-triggered average (STA) perform separate estimation of the linear and nonlinear components.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We know from [19] that this structure obtains provably sub-linear query time and provably subquadratic space, where the exponent depends on the probability of hash collisions for "similar" and "not-similar" sets.
Sentence 2: we also know that implementing MinHash with a poorly chosen hash function leads to constant bias in the estimation [28], and this constant then appears in the exponent of both the space and the query time of the search structure leading to worse theoretical guarantees.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When the constraints of (4) are relaxed using various convex norms, ||c − E Psrc(x) P (y|x) [f (X, Y )]|| ≤ �, the RBA classifier is obtained by � 1 -or � 2 -regularized maximum conditional likelihood estimation (Theorem 2) of the dual optimization problem [23,24], The regularization parameters in this approach can be chosen using straight-forward bounds on finite sampling error [24].
Sentence 2: the sample reweighted approach to learning under sample selection bias [1,7] also makes use of regularization [9], but appropriate regularization parameters for it must be haphazardly chosen based on how well the source samples represent the target data.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Nonlinear methods have been shown to find embeddings that nicely reflect the variability in the data caused by variation in face identity, pose, position, or lighting effects.
Sentence 2: it is not possible to tell these methods to extract a particular single factor for the purpose of, say intelligent image manipulation or pose identification, because the extracted factors are intermingled and may be represented simultaneously across all latent space dimensions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Under the assumption of independent and identically distributed (iid) Gaussian noise, the SFS problem can be formulated as a least squares problem in a space over three types of variables: the locations of the microphones, the locations of the acoustic events, and their emission times.
Sentence 2: this least squares problem is plagued by local minima, and the number of constraints is quite large.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: These forms of plasticity, generally termed Spike Timing Dependent Plasticity (STDP), increase the synaptic efficacy of a synapse when a presynaptic spike reaches the neuron a few milliseconds before the postsynaptic action potential.
Sentence 2: when the postsynaptic neuron fires immediately before the presynaptic neuron the strength of the synapse diminishes.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Such topic models with max-margin posterior constraints have shown superior performance in various settings [16,14,13,9].
Sentence 2: their constrained formulations, especially when using soft margin constraints for inseparable practical problems, make it infeasible or at least hard if possible at all 1 to directly apply Monte Carlo (MC) methods [10], which have been widely used in the posterior inference of likelihood based models, such as the collapsed Gibbs sampling methods for LDA [5].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To have fewer parameters, we set µ = µ i , i = 1, 2, 3 and µ i is replaced by µ in the following sections including experiments and supplementary materials.
Sentence 2: a typical iterative minimization process based on the alternating direction method of multipliers (ADMM) [15,16] can be written explicitly as minimizing the augmented Lagrangian function with respect to M i , L and E using ADMM is expensive in practice, and moreover, the global convergence can not be guaranteed.
Relationship: contrasting