sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: However, there is typically a large gap between the precision of distance computations in Hamming vs. product codes under the same bitrate, and ADC can be computed with similar speed ([2, 21], Section 4.4).
Sentence 2: given n samples from an underlying K-subspace mixture model that has been clustered correctly and an independently generated query q, with probability 1 , See Appendix B for a proof.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We modify an encoder-decoder translation model by adding a latent variable as a language agnostic representation which is encouraged to learn the meaning of the sentence.
Sentence 2: m-like step: Choose the words in y to maximize 1 S ∑ S s=1 log p(z (s) ) + log p(x|z (s) ) + log p θ (y|z (s) , x) using beam search.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In addition to the RSC condition, we assume that X has bounded column norms (specifically, n for all i = 1, . . . , p), and that the noise vector w ∈ R n has i.i.d.
Sentence 2: to motivate the need for a unified analysis, let us provide a brief (and hence necessarily incomplete) overview of the broad range of past and on-going work on high-dimensional inference.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In particular, the noise term ∆ T can be deterministic or even adversarial.
Sentence 2: ∆ T is weakly correlated with signal in the sense that ∆ T (v i , I, I) 2 = O(λ min /d) for all i ≤ k: in this case our noise condition reduces to ∆ T op = O(λ min / √ k), strictly improving over SVD initialization [2] in the "undercomplete" regime k = o(d).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The LATER model (Linear Approach to Threshold with Ergodic Rate), an instance of the additive offset model, incorporates prior probability as the starting point of a linearly rising decision variable and successfully predicts changes to saccade latency when discriminating between two low contrast stimuli [10].
Sentence 2: early in a trial, Π S has approximately the same width as in the neutral prior case, but it is shifted downwards to favor more sampling for d L (s < 0).
Relationship: neutral
Domain: NeurIPS
Sentence 1: It extends I-POMDP's belief update to larger model space, and therefore it serves as a generalized Bayesian learning method for multi-agent systems in which other agents' beliefs, transition, observation and reward functions are unknown.
Sentence 2: this additional resampling step is nontrivial, since empirically the samples diverge quickly due to the enormously enlarged sample space.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This method uses an information criterion to optimise the weights, under certain restrictions on Pr and Pr ′ (namely, Pr ′ must be known, while Pr can be either known exactly, Gaussian with unknown parameters, or approximated via kernel density estimation).
Sentence 2: since typically we only observe examples (x, y) drawn from Pr(x, y) rather than Pr ′ (x, y), we resort to computing the empirical average To avoid overfitting, instead of minimizing R emp directly we often minimize a regularized variant where Ω[θ] is a regularizer.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As ( µ1 τp ) 2 = O(1), gradient descent takes only a logarithmic number of iterations O(log 1 ϵ ) to converge now, which does not have the quadratic dependence on λ1 λ1−λ2 any more [Shamir, 2015, 2016a, Xu et al., 2017, Xu and Gao, 2018, Xu et al., 2018.
Sentence 2: we can also try other least-squares solvers, such as SVRG [Johnson and Zhang, 2013], accelerated SVRG [Garber et al., 2016], and coordinate descent .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The sequence consists of 59 greyscale frames, each of size 160 × 120 pixels (excluding the border).
Sentence 2: the system has detected not to use the previous frame at this point, but to revert to using the latent appearance image f . We have proposed a method for localising an object undergoing affine transform whilst simultaneously learning its shape and appearance.
Relationship: neutral
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: with the above assumption satisfied, let, P be a non-negative reference measure on the classifier space H known as the volumic measure.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , c k } ⊂ R p to be optimal for (RKM) are The interpretation of the set inclusion above is the following: for any center c l ∈ R p , every subgradient vector in ∂Φ(x i − c l ) must be a vector associated with a vector in P F λ (x i − c l ) (Theorem 10.13 in [19]).
Sentence 2: for all x, we have ∂Φ(x) ⊆ ∂Φ(x).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our logistic regression hypernym-only model trained on the newswire corpora has a 16% relative F-score improvement over the best WordNet classifier, while the combined hypernym/coordinate model has a 40% relative F-score improvement.
Sentence 2: for the second set of evaluations we hand-labeled a test set of 5,387 noun pairs from randomly-selected paragraphs within our corpus (with part-of-speech labels assigned by MINIPAR).
Relationship: neutral
Domain: NeurIPS
Sentence 1: When from the observations, a red node w ∈ V (r) is connected to at most a single green node, i.e., if v∈V (g) A vw ≤ 1, this red node is useless in the classification of green nodes.
Sentence 2: we present an algorithm with a memory requirement which depends on the density of the graph.
Relationship: neutral
Domain: NeurIPS
Sentence 1: It is easy to verify that indeed ∥v w ∥ 2 ≤ ∥β∥ 1 and for all x ∈ X , ⟨v w , ψ(x)⟩ = p(⟨w, x⟩).
Sentence 2: the problem is much harder in the agnostic case, namely, when L * γ > 0 and the distribution over examples can be arbitrary.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this criterion, the loss suffered for a training example (x (t) , y (t) ) is the negative log probability of the true parse according to the model, plus a regularization term: The convexity and differentiability of conditional log-likelihood ensure that gradient-based optimization procedures (e.g., conjugate gradient or L-BFGS [6]) will not converge to suboptimal local minima of the objective function.
Sentence 2: let X L denote an input space of all possible input sequences, and let Y L denote an output space of all possible output labels.
Relationship: neutral
Domain: NeurIPS
Sentence 1: ( 1) by P t and refer to it as the instantaneous primal problem to be solved on trial t. The dual optimization problem of P t is the maximization problem Each dual variable corresponds to a single constraint of the primal problem.
Sentence 2: the loss associated with each such violation is + and the loss of the categorizer is defined as the maximum over the losses induced by the violated pairs.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For the second-order term, it can be readily veri£ed that the Hessian of R(h) is block-diagonal with 2-by-2 blocks corresponding to h(x i ) and h(y i ) and 1-by-1 blocks for h(z i ).
Sentence 2: speci£cally, we want to rank a set of documents according to their relevance to a given query.
Relationship: neutral
Domain: NeurIPS
Sentence 1: An unbiased estimate of MMD 2 0 can be obtained using U -statistic where h(•) is the kernel of the U -statistic defined as h(x i , x j , y i , y Intuitively, the empirical test statistic MMD 2 u is expected to be small (close to zero) if P = Q, and large if P and Q are far apart.
Sentence 2: 1 and b grows on the order of p B, and this further simplifies our calculation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In each time step the learner chooses an action I t ∈ {1, . . . , K} and receives reward g It,t = µ i + η t where η t is the noise term that I assume to be sampled independently from a 1-subgaussian distribution that may depend on I t . This model subsumes both Gaussian and Bernoulli (or bounded) rewards.
Sentence 2: the order-optimal asymptotic regret can be significantly smaller.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In addition, it means that we cannot use standard minimax theory for Holder classes to establish lower bounds for the estimation error over GTF classes.
Sentence 2: let H d be the space spanned by all d-wise tensor products of falling factorial functions, i.e., for coefficients α ∈ R n (whose components we index by α i1,...,i d , for i 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Table 1: Percentages (mean and standard deviations) of correctly recognized single trials for the 2 motor imagination tasks for the 6 subjects and the average of them.
Sentence 2: this software, known for its zero localization error, was used as a localization tool to estimate the focus of intracranial activity at the occurrence of the four ErrP peaks described in Section 3.2.
Relationship: neutral
Domain: NeurIPS
Sentence 1: With a few exceptions, the experiments were conducted with δ = 50 and 4 computational threads.
Sentence 2: a similar idea was proposed in a-MMSB [5], using one parameter to determine inter-role link probabilities, rather than O(K 2 ) parameters for all pairs of distinct roles, as in the original MMSB [1].
Relationship: neutral
Domain: NeurIPS
Sentence 1: defined as the gradient of a potential field or as a gyroscopic "steering" force ϕ(x, O) = s(x, O) × v that effectively rotates the velocity vector [22] .
Sentence 2: as an example, consider a point-mass robot modeled as a double-integrator system with state x = (p, v) where p ∈ R d denotes position and v ∈ R d denotes velocity with d = 2 for planar workspaces and d = 3 for 3-D workspaces.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We write D E t as an abbreviation for D π E t . In a minor abuse of notation, we write s ∼ D π t to mean: draw state-action pair (s, a) ∼ D π t , and discard a.
Sentence 2: a sample (s, a) is drawn from D π t by first drawing s 1 ∼ α(•), then following policy π for time steps 1 through t, which generates a trajectory (s 1 , a 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The first term L adv ( Î, I) is the conditional adversarial loss defined by The second term L feature ( Î, I) is the feature matching loss [24].
Sentence 2: extension to style manipulation Although we focused on addition and deletion of objects as manipulation tasks, we can easily extend our framework to add control over object styles by conditioning the image generator with additional style vector s by G I ( M , I, s).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The objective function for φ given by , If φ is represented as a vector concatenating all of its parameters, the gradient of g(φ) takes the form For the quadratic nonlinearity, the gradients with respect to f (h t ) take the form ∂f Gradients for the quadratic model with multiplicative interactions take a similar form.
Sentence 2: we set the dynamics matrix A by fitting the model to a single neuron recording from V1 [4].
Relationship: neutral
Domain: NeurIPS
Sentence 1: In fact, at the beginning tUCB selects arms according to a UCB strategy, since no prior information about the models ⇥ is available.
Sentence 2: in fact, if J is large, it could be preferable to run a model identification algorithm instead of umUCB in earlier episodes so as to improve the quality of the estimates μi (✓).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Moreover, we note that in this scenario, the standard choice of ν = 0.5 in Theorem 2 is inferior to choosing smaller parameter values of ν = 0.1 or ν = 0.001.
Sentence 2: the constant before σ 2 n ln 1 q k * δ approaches 8 when → ∞ (with default ν = 0.5), which is smaller than the constant of Theorem 1 which is about 30.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The representation of a neuron v is measured by the activation vector [Raghu et al., 2017] of the neuron v over the d inputs, For any subset X ⊆ X , we denote the vector set {z x : x ∈ X} by z X for short.
Sentence 2: again, there are few simple matches with small sizes.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In order to learn useful, predictive representations in a fully unsupervised setting, we apply a graph-based loss function to the output representations, z u , ∀u ∈ V, and tune the weight matrices, W k , ∀k ∈ {1, ..., K}, and parameters of the aggregator functions via stochastic gradient descent.
Sentence 2: we can derive an inductive variant of the GCN approach by replacing lines 4 and 5 in Algorithm 1 with the following: 4 We call this modified mean-based aggregator convolutional since it is a rough, linear approximation of a localized spectral convolution [17].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Also, we still do not know the lower bound of the utility using this measure.
Sentence 2: suppose the loss function f (x, z) is G-Lipschitz , and L-smooth over x ∈ C . In DP-AccMD, let σ be as in (9) and w be a function that is 1-strongly convex with respect to || • || C . Then if we have The total gradient complexity is O .
Relationship: neutral
Domain: NeurIPS
Sentence 1: This results in the desirable behavior of the labels varying smoothly over the graph, as sought by, e.g., spectral clustering approaches [2], diffusion kernels [5], and the Gaussian random field approach [9].
Sentence 2: informally, a smooth eigenvector has the property that two elements of the vector have similar values if there are many large weight paths between them on the graph.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then let I m = {i r i = 1, i ≤ m} and J m = {j t j = 1, j ≤ m} and let I = I ∞ and J = J ∞ .
Sentence 2: the rewards in Tetris are received sparsely, unlike the approximation error, which makes the value function less sensitive to the discount factor than the approximation error.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then ξ b st (Λ), in addition to being twice the variance of the Bernoulli indicator of the edge (s, t), has the following nice interpretation: For each pair of graphs, we can ask how often they disagree on the presence of the edge: ξ b st (Λ) is the fraction of times they disagree.
Sentence 2: we want to "overselect" instead of "underselect".
Relationship: neutral
Domain: NeurIPS
Sentence 1: This corresponds to insulating the boundary, also known as the Neumann boundary condition.
Sentence 2: using the group lasso penalty [18], we estimate the significance of the subnetwork by solving the following optimization problem 2 where J is the total number of subnetworks considered and X ( j) and β ( j) denote the columns of X , and entries of β corresponding to the subnetwork j, respectively.
Relationship: neutral
Domain: NeurIPS
Sentence 1: So the leave-one-out hyperparameter learning will push 1/σ h to zero and 1/σ v to infinity, i.e., all points can transfer only horizontally.
Sentence 2: the difference between LOOHL+CMN and LOOHL+Thrd, compared with the difference between 5-CV+CMN and 5-CV+Thrd, can be viewed as an approximate indicator of how well the graph is learned by LOOHL.
Relationship: neutral
Domain: NeurIPS
Sentence 1: There is a closed lower dimensional set E ⊂ L such that the function τ * is locally constant on the complement of E. Theorem 1 implies that, with probability 1, if the model estimate m and the true model m are sufficiently close, then the actions recommended by the exactly optimal policies for both models are identical.
Sentence 2: by bounding the amount of time allowed between successive screenings.
Relationship: neutral
Domain: NeurIPS
Sentence 1: At last, this method is complicated and slow because too much calculation is involved.
Sentence 2: we can find an expansion for w k in the form Using this expansion and the definition of m Φ i , we have Replacing the dot-product by some kernel function k(x, y) and defining l . Similarly, we have Similarly, one can simplify W T S Φ w W. First, we have w T k (Φ(x α T k S i (e j e T j − j ), I ni×ni is the n i × n i identity matrix, 1 ni is the n i -dimensional vector of 1's, and e j is the canonical basis vector of n i dimensions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For a particular diagonal line, one random vector zeros out a set of approximately d/2 entries, and the second random vector zeros out another set of d/2 entries, chosen randomly and independently of the first set.
Sentence 2: as depicted in Figure 1 (middle), HOLEX visually first forms the full tensor product of a and b, then multiplies each row with the corresponding dimension in c, and finally sums up along each (circular) diagonal line.
Relationship: neutral
Domain: NeurIPS
Sentence 1: On the computational front, DPMBQ costs O(n 3 ).
Sentence 2: next, the GP is conditioned on the information in {(x i , f . Formulae for the mean function m n : Ω → R and covariance function k n : Ω × Ω → R are standard can be found in [31,Eqns.
Relationship: neutral
Domain: NeurIPS
Sentence 1: And, by definition of πk , λk is the gain of the chosen policy πk for MDP Mk .
Sentence 2: the rounds t = 1, . . . , T are broken into consecutive epochs as follows: the k th epoch begins at the round τ k immediately after the end of (k − 1) th epoch and ends at the first round τ such that for some state-action pair s, a, N τ s,a ≥ 2N τ k s,a .
Relationship: neutral
Domain: NeurIPS
Sentence 1: (Right:) Fraction of samples that RMSE is smaller than 10 −7 , among 100 samples of rank-3 data matrix U V T of size 1000 × 1000, with the entries of U and V drawn from a Gaussian distribution of mean 0 and unit variance.
Sentence 2: we see that if the topology is not locally-tree-like but with some noise, for example with some additional cliques, both trimming of the data matrix and Bethe Hessian perform much worse, reporting a wrong rank, and giving a large reconstruction error, as illustrated in Fig.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the two extreme cases, we know that the "sample selecting model" constraint I v saturates when nk ⌧ mp (e.g., k = 1 and p = n), and the "model selecting sample" constraint I u saturates when nk mp (e.g., k = m and p = 1).
Sentence 2: these observations suggest we start at the first regime mp  nk with small p and large k, and gradually switch to the second regime with mp nk by slowly increasing p and decreasing k. In earlier stages, we also should set the diversity weights and to be large, and then slowly reduce them as we move towards the second regime.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 4 This set of bootstrapped examples D L is then fed as training data into a supervised or semi-supervised algorithm Φ(D L , D U ), and MFA outputs the resulting classifier.
Sentence 2: p (y = 1|x i = r) > P (y = 1|x i = r ) if r > r . • The monotonicity is non-trivial in that P (x i ) has positive probability for more than one feature value.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The proof follows immediately from the strict convexity of mutual information [7] and the fact that the two conditions guarantee that each Q(y|x) appears non-trivially in at least one mutual information term.
Sentence 2: we begin by rewriting each mutual information term I R (x; y) in the criterion = min where the variational distribution Q R (y) can be chosen independently from Q(y|x) but the unique minimum is attained when We can extend the regularizer over both {Q(y|x)} and {Q R (y)} by defining The local propagation algorithm follows from optimizing each Q(y|x) based on fixed {Q R (y)} and subsequently finding each Q R (y) given fixed {Q(y|x)}.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Focusing on the new comparators only, observe that their progress is nearly statistically equivalent during the first 20 iterations.
Sentence 2: define f n min = min i=1,...,n y i , the smallest y-value seen so far, and let I(x) = max{0, f n min − Y (x)} be the improvement at x. I(x) is largest when Y (x) ∼ f n (x) has substantial distribution below f n min .
Relationship: neutral
Domain: NeurIPS
Sentence 1: An analysis of IT neurons that responded to the target stimulus showed that the average size of the receptive fields shrinks from approximately 56 degrees in a blank background to approximately 12 degrees with a complex scene [8].
Sentence 2: the second object, a distractor object, was placed on the opposite side of the fovea at a fixed distance of ¢ degrees from the fovea.
Relationship: neutral
Domain: NeurIPS
Sentence 1: It produced an average fitness within 1% of the optimum for (): between 0.1 and 0.4 and L = 32, 64,128,256 over a 100 trials, but for learning rates below 0.1 and L = 256 the average fitness fell as low as 4% below optimum.
Sentence 2: one might think that the needle-in-the-haystack problem is hard in a special way, and results on this problem are not relevant to other problems.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Moreover, we also propose a local method for multi-task regression problems.
Sentence 2: our SMO algorithm updates a violating pair at one step by choosing the most violating pair {j, k} with j and k defined as j = arg max l∈S1∪S2 {w i a l + b l } and k = arg min l∈S2∪S3∪S4 {w i a l + b l }.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For example, from a Mondrian process on products of intervals, we can construct a Mondrian process on all of R D . Note that if the domains have infinite measure, the tree of cuts will be infinitely deep with no root and infinitely many leaves (being the infinite partition of the product space).
Sentence 2: ( 11) is the product of two stick-breaking priors.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A primary insight is that, since Algorithm 3 regularizes with the L 2 norm and penalizes with hinge-loss, omitting the positive semidefinite requirement for M and vectorizing M makes the algorithm equivalent to a oneclass, linear support vector machine with O(n 3 ) input vectors.
Sentence 2: for maximum weight subgraph connectivity (e.g., b-matching), we use a cutting-plane method [10], iteratively finding the worst violating constraint and adding it to a working-set.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Namely, for any observation signal y, and any nonzero scalar α and integer shift τ , the following equality always holds Here, s −τ [v] denotes the cyclic shift of the vector v by τ entries: Clearly, both scaling and cyclic shifts preserve the short-and-sparse structure of (a 0 , x 0 ).
Sentence 2: note that if we could recover some shift s τ [x 0 ], we could subsequently determine s −τ [a 0 ] by solving a linear system of equations, and hence solve the deconvolution problem, up to the shift ambiguity.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Under a mild technical assumption, Proposition 9 suggests that losing connections monotonically increases the MSD, and individuals tend to maintain their friendships to obtain a lower MSD as a global objective.
Sentence 2: essentially, if there is no connection between agents i and j, for < min{p ii , p jj }, corresponds to a new communication matrix adding the edge {i, j} with a weight to the network G, and subtracting from self-reliance of agents i and j.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In Figure 1(b), a link is predicted between the green and yellow atoms.
Sentence 2: we design a graph generation procedure that can be formulated as a Markov Decision Process (MDP), which requires the state transition dynamics to satisfy the Markov property: p(s t+1 |s t , ...s 0 ) = p(s t+1 |s t ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We compare to two very recent approaches to Bayesian neural networks, namely an approach based on expectation propagation [HLA15] and an approach based on variational Bayes [BCKW15].
Sentence 2: we first compute the posterior distribution over the model parameters, p(θ|D N ) ∝ p(θ) N i=1 p(y i |x i , θ), where D N = {(x i , y i )} N i=1 , x i ∈ X D is the i'th input (where D is the number of features), and y i ∈ Y is the i'th output.
Relationship: neutral