sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Therefore, maximizing SNR ẑ j is equivalent to Due to the use of batch normalization, Eq.
Sentence 2: ( 12) is a constant with respect to each sample of z s j , and thus we have ∂ E ẑs j 2 /∂z s j (m) = 0, ∀m ∈ B, where B denotes a set of mini-batch samples, and z s j (m) denotes the value of z s j corresponding to sample m. we can now remove the feature correlation effect of multiplicative noise without causing optimization difficulties.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [1] have shown that the error rate on unseen test samples can be upper bounded by the disagreement between the classification-decisions obtained from independent characterizations (i.e., views) of the data.
Sentence 2: in the web-page example, misclassification rate can be indirectly minimized by reducing the rate of disagreement between hyperlink-based and content-based classifiers, provided these characterizations are independent conditional on the class.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the independent cascade model, a node a influences each of its neighbors b with some probability q ab , independently.
Sentence 2: given a seed set of nodes S, the set of nodes influenced is the number of nodes connected to some node in S in the random subgraph of the network which contains every edge ab independently with probability q ab .We define f (S) to be the expected number of nodes influenced by S according to the independent cascade model over some weighted social network.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Put another way, a rank value y induces the vector (Yl, ... , Yk-d = (+1, ... , +1 , -1 , ... , -1) where the maximal index r for which Yr = +1 is y-1.
Sentence 2: the prediction of a ranking rule is correct if Yr(w• x -br ) > 0 for all r. If the algorithm makes a mistake by ranking x as fj instead of Y then there is at least one threshold, indexed r, for which the value of W• x is on the wrong side of br , i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the analysis of neural responses, once we fix our time resolution the response becomes a set of discrete "words," and estimates of the information content in the response are de-termined by the probability distribution on this discrete space.
Sentence 2: what all of these examples have in common is that we often need to draw some conclusions with data sets that are not in the asymptotic limit N K. while we might use a large corpus to sample the distribution of words in English by brute force (reaching N K with K the size of the vocabulary), we can hardly do the same for three or four word phrases.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: process is completely characterized by P x n = p, hence this problem coincides with the previous one.
Sentence 2: for Markov chains, we seek to estimate the transition matrix M . instead of producing a probability distribution Px n , the estimator M maps every sequence For two distributions p and q over [k], let L(p, q) be the loss when p is approximated by q.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Step 2, although the sampling process terminates when a solved node n is reached, we associate every configuration x >n of X >n with probability g(x >n |x ≤n )/v n which is appropriate in lieu of (1).
Sentence 2: we can show that this two-step sampling scheme induces a proposal distribution, denoted q S (x), which can be expressed as: where AND(T x ∩ S) is the set of all AND nodes of the partial solution tree T x ∩ S. By applying (3), and noticing that the upper bound is the initial heuristic for any node in OPEN(S) and is exact at any solved node, we re-write q S (x) as q S (x) actually provides bounded importance weights that can use the refined upper bound U S : Proposition 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this subsection, we consider the setting where all the elements are observed (with noise) and the goal is to recover the underlying low-rank tensor without noise.
Sentence 2: since all the elements are observed only once, X is simply a vectorization (M = N ), and the lefthand side of inequality (10) gives the quantity of interest ∥X(∆)∥ 2 2 = Ŵ − W * F . the remaining task is to bound X * (ϵ) mean as in the following lemma.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The gradient is Note that, for graph matching problem, we have W T = W. Thus, the second derivative is Therefore, Φ(x, x) is a concave function in x and has a unique global maximum.
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: reasoning
Domain: NeurIPS
Sentence 1: In each iteration, it is allowed to gather information by asking for a single pairwise comparison between two items-or, using the dueling bandits jargon, to pull two arms.
Sentence 2: it selects two items i and j, and then observes either preference i j or j i; the former occurs with probability p i,j as defined in (1), the latter with probability p j,i = 1 p i,j .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As the temporal structure of the auditory receptive fields was less dominant than their spectral structure (Figure 2C), we focused solely on the spectral domain and did not attempt to address temporal information.
Sentence 2: the inputs for the new model (n = 100,000) were short-time frequency spectra of 128 pixels each (24 pixels = 1 octave).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the simulations we let (µ 1 , σ 1 ) = (−2, 1), (µ 2 , σ 2 ) = (2, 1), σ x = 0.1, and c = 20, and p = 1000.
Sentence 2: f Y |X is a highly nonlinear function of x, and even η, and x is high-dimensional.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the proposed framework, modal decomposition is first obtained in terms of learned observables g; thus, the values of g must be back-projected to the space of the original measurements y to obtain a physically meaningful representation of the dynamic modes.
Sentence 2: we modify the loss function by employing an additional term such that the original measurements y can be reconstructed from the values of g by a reconstructor h, i.e., y ≈ h(g( x)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Learning of bottom-up connections (i.e., feedforward weights) is driven by such error signals through local synaptic plasticity.
Sentence 2: in contrast to previous approaches (Marblestone et al., 2016), in our framework a given neuron is used simultaneously for activity propagation (at the somatic level), error encoding (at distal dendrites) and error propagation to the soma without the need for separate phases.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically we define this bump function in the transformed space as where r is the radius from the origin, r e is the maximum radius of any point generated in the initial scaling or subsequent evaluations, and r ∞ is a parameter set to 1.5r e by default.
Sentence 2: the acquisition function also decays and new points are never suggested arbitrarily far away.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because Υ(c, m) is entirely dependent on the two matrices Υ d and Υ e as well as our kernel matrix, we can calculate an estimate for any set of nodes in the graph by explicitly calculating only those rows of Υ which correspond to those nodes and then simply multiplying that sub-matrix by the vector y.
Sentence 2: if we have an efficient way to compute arbitrary entries of the kernel matrix K 1 , we can estimate functions anywhere in the graph.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let (Ω, d Ω ) be a compact metric space, B(Ω) be its Borel σ-algebra, and X := M 1 (Ω) be the set of all Borel probability measures on Ω.
Sentence 2: then the topology describing weak convergence of probability measures can be metrized, e.g., by the Prohorov metric d X (P, P ) := inf ε > 0 : P(A) ≤ P (A ε ) + ε for all A ∈ B(Ω) , P, P ∈ X , where  the map ρ : M 1 (Ω) → H Ω satisfies the assumptions of Theorem 2.2, and hence the Gaussian-type RBF kernel k σ (P, P ) : is universal and obviously bounded.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: More generally, functions of the form f (λ) = λ −α , α ≥ 0, allow for combining shared features and task-specific features to some degree tuned by the exponent α.
Sentence 2: moreover, the regularizer (2.2) ensures that the optimal representation (optimal D) is a function of the tasks' covariance W W . we propose to solve the minimization problem for functions f belonging to an appropriate class.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using the Chebyshev interpolant of log(1 + αx), we approximate log | K| by For stochastic estimation of tr(T j (B)), we only need to compute z T T j (B)z for each given probe vector z.
Sentence 2: we compute vectors w j = T j (B)z and ∂w j /∂θ i via the coupled recurrences This gives the estimators each derivative of the approximation costs two extra MVMs per term.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, by scanning the solution regions from left to right according to the sorted s values, and maintaining some intermediate computed variables, we are able to find the optimal solution, as summarized by the following lemma: Lemma 1.
Sentence 2: the subproblem (5) is convex with respect to d and can be solved exactly in O( P 0 log( P 0 )), where P 0 is the number of nonzero elements in tensor P. we can efficiently update d since P is a very sparse tensor with only a small number of nonzero elements.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 2 implies that γ m in ( 12) satisfy the monotonicity condition: γ 1 ≥ γ 2 ≥ .
Sentence 2: . . ≥ γ M . equations ( 12) constitute the parametric submodular minimization problem as defined above, which reduces to the monotonic parametric max-flow problem for pairwise E. Let • denote the largest integer not exceeding an argument of the operation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Figures 2 (c) and (d) show that the responses of two groups of participants who answered the arithmetic problem in (a) and (b) differed as predicted (χ 2 (1) = 5.208, p < 0.05, using Yates' chi-square correction).
Sentence 2: asking participants to solve arithmetic problems is an effective method for testing reference frame inference and perceived orientations can influence higher level cognition.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let T unf olded ∈ R m×n 2 be the unfolded tensor T such that T unf olded (:, (j 1 − 1)n + j 2 ) = T (j 1 , j 2 , :) for 1 ≤ j 1 , j 2 ≤ n. Without noise, the right singular vector of this matrix is u 1 ⊗ w 1 which corresponds to the singular value σ 1 .
Sentence 2: selecting k 2 indices of this singular vector with largest magnitudes would recover the index set J 1 × J 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (2) The gradients of L(δ, w) w.r.t.
Sentence 2: the weights {w i , w α i } are marginal and conditional entropies defined on the beliefs {µ i , µ α }, respectively, the optimal weights should satisfy the following KKT condition where Hi = w i H(x i ; µ i ) + α w α i H(x i |x i+1:c ; µ α ) is the (weighted) average entropy on node i.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In a linear factor model, a user's rating of an item is modeled as a linear combination of these factors, with user-specific coefficients and item-specific factor values.
Sentence 2: given a N × M preference matrix for N users and M items, a K-factor model fits it with a N × K coefficient matrix U and a M × K factor matrix V as U V . Various computational methods have been successfully developed to implement such an idea, including probabilistic matrix factorization (PMF) [13,12] and deterministic reconstruction/approximation error minimization, e.g., max-margin matrix factorization (M 3 F) with hinge loss [14,11,16].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any T 1 ⊂ T 2 and any given example x / ∈ T 2 , we have Similarly, we have g(T 2 ∪ x) − g(T 2 ) = wq x r x − S x,x r 2 x − 2r x j∈T2 S x,j r j .
Sentence 2: we have which completes the proof of (P1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is implemented by training on examples available for nodes in the current subtree (i.e., the tasks below the current node), while similarity to the parent classifier is induced through regularization.
Sentence 2: for each node n, one solves the following optimization problem, where p is the parent node of n (with the special case of w * p = 0 for the root node), is an appropriate loss function (e.g., the hinge-loss).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, a stratification-based approach suffers in estimation when the count distributions of the point processes under consideration are flat, since in this situation the spike train realizations tend to exist in separate Euclidean partitions, and given a finite set of realizations, it becomes difficult to populate each partition sufficiently.
Sentence 2: other methods should be investigated that allow two spike trains to interact irrespective of their spike counts.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [3] is only for the special case when k = d. We extend their result to every k and d. In particular, we show that the sample complexity for collaborative learning has to be Ω(max{d ln k, k ln d}/ ) for constant δ (Theorem 6).
Sentence 2: the sample complexity of our algorithm is optimal when k = d O(1) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Unfortunately, solving a classification problem with a bounded loss function is computationally intractable, because the problem is no longer convex and even state-of-the-art solvers like, for example, CPLEX [20] fail to found an exact solution, when the training set size exceeds few tens of samples.
Sentence 2: we propose here to find an approximate solution through well-known algorithms like, for example, the Peeling [6] or the Convex-Concave Constrained Programming (CCCP) technique [14,21,22].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, the f -spectral distance can be expressed as S f (x, y) = ||Ψ(x) − Ψ(y)|| 2 2 , where Ψ(x) is the x th row of Ψ.
Sentence 2: ψ represents an Euclidean embedding of G where each node x is represented by the vector Ψ(x).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, 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).
Sentence 2: we prove that SGD has a strong inductive bias on structured data: Instead of finding a bad global optima that can fit arbitrary labels, it actually finds those with good generalization guarantees.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is straightforward to see that, on the same event, we havek ŵmle wk 2  C p d/n for some absolute constant C > 0.
Sentence 2: the necessary and sufficient conditions on SNR for approximate recovery of w lie between C 0 d/ log log(n) and n C 00 (for absolute constants C 0 , C 00 > 0).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Mixture models constitute some of the most popular models for MFVB application [1,2] and are often used as an example of where MFVB covariance estimates may go awry [5,6].
Sentence 2: we will consider in detail a Gaussian mixture model (GMM) consisting of a K-component mixture of P -dimensional multivariate normals with unknown component means, covariances, and weights.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, for an uninfected j ∈ [n], to compute the corresponding row of Y , y T j ∈ R k , we only need k independent linear equations.
Sentence 2: if M is non-degenerate, to compute y T j we only need k entries of the j'th column of M with row indices in L. Casting the condition in terms of the sampling graph G Ω , y T j can be computed and vertex v j ∈ V be marked as infected if there are at least k edges from v j to infected vertices in L. Analogously, x T i can be computed and the vertex u i ∈ U be marked as infected if there are at least k edges from u i to previously infected vertices R. Observe that M = XY T = XW W −1 Y T , for any invertible matrix W ∈ R k×k .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For an RKHS based on a Gaussian kernel, the entropy condition (10) holds for any γ > 0 [23].
Sentence 2: furthermore, (9) trivially holds via the Cauchy-Schwarz inequality: by Theorem 2(a), ŵ H = ĝn H = O P ( g 0 H ), so the penalty term λ n ŵ 2 vanishes at the same rate as λ n . We have arrived at the following estimator for the KL divergence: log G is an RKHS.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The only clear discrepancy between the model predictions and the data is the cluster of points at the lower left, labeled as Decisions 6-13 in Figure 2a.
Sentence 2: these are all cases in which effect x appears in all options and therefore these decisions provide no information about a decision maker's preference for x. the models assign the same ranking to this group as to the group of decisions in which there is only a single option (Decisions 1-5).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Assuming uniformly distributed variables z n and independence between z n and x n implies that w T k x n + b k = 0.
Sentence 2: using the second-order expansion of the log-sum-exp function ϕ(u) = log( K k=1 exp(u k )) around 0 leads to the following approximation of the terms depending on (w, b): where K is the usual centering projection matrix, and X = (x 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a given f and a radius f ∅ = f (y) this surface area is given by the equation where I denotes the set of all multi-indices of inner nodes, n I the number of leaves of the subtree under I and B [a, b] the beta function.
Sentence 2: if the partition function of the radial distribution can be computed easily, so can the partition function of the multivariate L p -nested distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Mondrian forests excel in this setting: at iteration N + 1, each Mondrian tree T ⇠ MT ( , D 1:N ) is updated to incorporate the next labeled example (x N +1 , y N +1 ) by sampling an extended tree T 0 from a distribution MTx( , T, D N +1 ).
Sentence 2: using properties of the Mondrian process, we can choose a probability distribution MTx such that T 0 = T on D 1:N and T 0 is distributed according to MT ( , D 1:N +1 ), i.e., the distribution of Mondrian trees trained on a dataset in an incremental fashion is the same as that of Mondrian trees trained on the same dataset in a batch fashion, irrespective of the order in which the data points are observed.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We denote by Û (π) the utility of the policy π in an MDP whose first-order transition probabilities are given by P θ (and similarly V π the value function in the same MDP).
Sentence 2: we have 2 for all π, then finding the optimal policy in the estimated MDP that uses parameters P θ (using value iteration or any other algorithm) will give a policy whose utility is within 2ε of the optimal utility.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any ε ∈ (0, 1] and µ ∈ (0, 1 The complete proof of Theorem 2 is given in the supplementary materials.
Sentence 2: the main idea is to show that ρ z is bounded from below by a positive constant with high probability, and then bound the excess misclassification error R(sign choosing C such that lim n→∞ C/n = 0, which is equivalent to lim n→∞ (1 + C κ,H ) 2 /n = 0, then R(sign(f z )) can be arbitrarily close to the Bayes error R * , with high probability, as long as n is sufficiently large.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To solve this optimization problem, we introduce the Accelerated Parallel ProximaL gradiEnT (APPLET) method, which tackles the non-smoothness of overlapping-group-lasso penalty via proximal gradient [20,9], and the huge number of training samples by Map-Reduce parallel computing [10].
Sentence 2: the contributions made in this paper are: (1) We incorporate the semantic relationships between object classes, into an augmented multi-class logistic regression formulation, regularized by the overlapping-group-lasso penalty.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: where the homogeneity measure among nodes in e, Hom w (x, e), is also the inner product of the parameter vector w and the feature vector ϕ e (x) and takes values of both signs such that a large positive value means strong homogeneity while a large negative value means high degree of nonhomogeneity.
Sentence 2: note that the proposed discriminant function for higher-order correlation clustering is decomposed into two terms by assigning different parameter vectors to the pairwise edge set E p and the higher-order edge set E h such that w = [w T p , w T h ] T . in addition to the pairwise similarity between neighboring superpixels, the proposed higher-order correlation clustering considers a broad homogeneous region reflecting higher-order relations among superpixels.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We generate four hard negatives: very small in size relative to the foreground, the resulting hard negatives can have high overlap with the foreground, which will introduce undesirable noise (false negatives) when training the detector.
Sentence 2: we shrink any hard negative that overlaps with the foreground estimate by more than 50%, until its overlap is 50% (we adjust the boundary that does not coincide with any of the foreground estimation boundaries).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (b) The dependence of the upper bound on neural networks in [3] is characterized by the total number of parameters of neural networks, which can be quite loose by considering recent work, e.g., [20,27].
Sentence 2: the goal of this paper is to address the above issue (a) by developing a lower bound on the minimax estimation error of d Fnn (µ, ν) (see Section 2.2 for the precise formulation) and to address issue (b) by developing a tighter upper bound than [3].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now each steganographic algorithm has an associated training  Due to the non-convexity of the models in the training scheme, we cannot guarantee that two separate parties training on the same images will converge to the same model weights, and so learn the same embedding and decoding algorithms.
Sentence 2: prior to steganographic communication, we require one of the communicating parties to train the scheme locally, encrypt model information and pass it to the other party along with information about the set of training images.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Second, a previous related study [8] has shown that, when considering large neuronal populations with a uniform distribution of stimulus preferences (such as an hypercolumn in V1 containing all stimulus orientations) the tuning width of individual neurons which is optimal for population coding depends crucially on the number of stimulus features being encoded.
Sentence 2: it is interesting to investigate how the optimal arrangement of corticocortical synaptic systems depends on the number of stimulus features being encoded.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our first task is to compute the coefficients P in the objective function.
Sentence 2: note that, P ( 0 H P P FQ #B &P FQ s( 0 © P P Q W P Q , as basis B has scope restricted to S . Here, P P Q represents the marginal of the state relevance weights P over S s . the coefficients P can be pre-computed efficiently if P is represented compactly by its marginals P P S c pQ .
Relationship: reasoning