sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Note that in the classical linear regression model where y i = w> x i + " i for i 2 [n], the maximum likelihood estimator ŵmle satisfies Ek ŵmle wk 2  C p d/n, where C > 0 is an absolute constant.
Sentence 2: the SNR requirement to approximately recover w up to (say) Euclidean distancek wk 2 /24 is SNR 24 2 Cd/n.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Although this decomposition restricts the covariance matrix representation, it dramatically reduces the number of variational parameters in the covariance matrix from M 2 X M 2 H to M 2 X + M 2 H . Thanks to the above decomposition, the lower bound can be rearranged to speed up the computation, Similarly, the KL-divergence between q(U) and p(U) can also take advantage of the above decomposition: As shown in the above equations, the direct computation of Kronecker products is completely avoided.
Sentence 2: the computational complexity of the lower bound is reduced to , which is comparable to the complexity of sparse GPs with independent observations O(N M max(D, M )).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The probability model can be written as where We defined s, r, and ψ as Then, the estimating function is given by where we used To calculate the conditional expectation of log T 1 , let us use Bayes's Theorem: By transforming random variables, (T 1 , T 2 , T 3 , ..., T m ), into (s, T 2 , T 3 , ..., T m ), we have where the beta function is defined as Similarly, we have where the digamma function is defined as Note that E[log(T 1 )|s] does not depend on the unknown function, k(ξ).
Sentence 2: we have The form of u I can be understood as follows.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: we assume (10) is true for k. Using Eq.
Sentence 2: ( 11), we have From above equation and ( 8) with x = y k , we get On the other hand, from the definition of the estimate sequence and (8), from these two equations, we have Since g is Lipschitz smooth, we bound f (x k+1 ) as follows: Using ( 16), ( 17), (12), and x k+1 − y k = −η(v k + ξ k ) we have where for the first inequality we used Here, we give the following where for the first inequality we used ) and convexity of g and h. Thus we have By taking expectation with respect to the history of random variables I 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Taking the high probability statements of Lemma 3 and Lemma 4 with /2 2 [0, 1/e] tells us that with probability at least 1 , kw which is required for Lemma 4 to hold.
Sentence 2: if we set the algorithm parameter ✏ as indicated in the statement of theorem, we have that with probability at least 1 T 1/2 ↵ for all t 2 {T ↵ + 1, . . . , T } that a t = 1 and v t p t  ✏, which follows from the same argument used for Lemma 6.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, we introduce an additional tuning parameter µ > 0 and look for minimizing the criterion If we make the change of variables φ 1 = Mβ/σ, φ 2 = Nβ/σ and ρ = 1/σ, we get a convex function for which the first-order conditions [20] take the form Thus, any minimizer of (2) should satisfy these conditions.
Sentence 2: to simplify the problem of optimization we propose to replace minimization of (2) by the minimization of the weighted 1norm j ωj |(Mβ) j | subject to some constraints that are as close as possible to (3)(4)(5).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is because, if not, then, using Theorem 6, in each of the good major cycles t * + 1, t * + 2, . . . t * + t , err(x) falls additively by > ε 4 /8Q 2 and thus err(x t * +t ) < err(x t * ) − ε 2 ≤ 0, which is a contradiction.
Sentence 2: in O(Q 2 /ε 2 ) good major cycles, the algorithm obtains an x = x t * * with ∆(x) ≤ ε 2 , proving Theorem 4.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The remaining terms correspond to the comparisons between a non-winner arm and the remaining arms, and is bounded by O K 2 log log T . As demonstrated in [6], L C is relatively small compared to K, and can be viewed as a constant.
Sentence 2: the total regret In particular, this asymptotic trend can be easily seen for Condorcet dueling bandits where L C = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By definition, an execution graph G for an execution contains a vertex if its label is a substring of the execution's trace t. Likewise, an edge is contained if an aggregation of the vertex labels is a substring of t. It follows 1 that the predictive distribution can be reformulated as in Equation 2to predict the probability of seeing the code position s = s n after a fragment of preceding statements f = s 1 . . . s n−1 using the trace representation of an execution.
Sentence 2: it is not neccessary to represent execution graphs G explicitly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 1) is thus equivalent to min Using the change of variable βv = v∈V , this implies that given the optimal η (and associated ζ), β corresponds to the solution of the regular supervised learning problem with kernel matrix K = w∈V ζ w K w , where K w is n × n the kernel matrix associated with kernel k w . Moreover, the solution is then β w = ζ w n i=1 α i Φ w (x i ), where α ∈ R n are the dual parameters associated with the single kernel learning problem.
Sentence 2: the solution is entirely determined by α ∈ R n and η ∈ R V (and its corresponding ζ ∈ R V ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here we use the standard squared exponential (SE) kernel.
Sentence 2: x ∼ N (µ1, Σ), where Σ is the positive definite covariance matrix defined by For notational convenience, we define the hyperparameter set θ = [µ; γ; κ; σ 2 f ; σ 2 v ].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, two images containing one common object may also have other visually incompatible objects, which makes these two images less similar in terms of the B2B distance.
Sentence 2: instead of measuring the similarities between bags, in this paper we approach MIL from a new perspective using the Class-to-Bag (C2B) distance, which assesses the relationships between the classes and the bags.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 5) can be viewed as a data labelling problem, where each value x j or y j takes a label in [1, N ], with a monotonicity constraint.
Sentence 2: let us now consider the graph (V, E) where nodes V = {v t } t=1...2M represent the list of all x j and y j in a sorted order and the edges E = {(v s , v t )} connect all pairs (x j , y j ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Suppose the initial values µ x , µ y satisfy µ x = ϕ X L1(fµ y ) .
Sentence 2: then, at every iteration t ≥ 1 of EGT, the corresponding solution z t = [x t ; y t ] satisfies x t ∈ X , y t ∈ Y, the excessive gap condition is maintained, and EGT has a convergence rate of O( 1 T ) [24].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that these geometrybased methods require high-resolution images at high frame rates.
Sentence 2: we sample high-resolution images (1024 × 1024 vs. 224 × 224 for our policies) at 5× the frame rate (145 vs. 30 images for our policy for a trajectory of length 30) along the reference trajectory along with ground truth poses.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We now recall that the individual rectangles of the class "m x n" differ only in their positions within the 4 x 4 grid.
Sentence 2: the cardinality of the class "m x n" is equal to the number ofunique ways N m x n in which an m x n or n x m rectangle can be selected from a 4 x 4 grid, where Thus, the probability associated with an individual rectangle of class "m x n" is P N (m x n) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Pair-based models with all-to-all interaction [4, 5, 6, 7, 8, 9, 10] yield under Poisson stimulation (protocol P2) a total weight change that is linear in presynaptic and postsynaptic frequencies.
Sentence 2: as a function of postsynaptic frequency we always find a straight line with a slope that depends on the integral of the STDP function [5,7].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let I n denote an indicator vector of length K that records which signatures appear at node n. For the root node, all K signatures are present, so we have I n = [1, . . . , 1].
Sentence 2: following the typical random forest protocol [2], the training instances are recursively split according to a randomized test; it compares one dimension of the signature against a threshold t, then propagates each one to the left child l or right child r depending on the outcome, yielding indicator vectors I l and I r . Specifically, if during training we must choose two things at each node: the query attribute m and the threshold t, represented jointly as the split (m, t).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: and the encoding functions i (θ) are of the finite-dimensional form considered above, i (θ) = K T i Φ(θ), then the log f [b i + i (ϑ)] term in the definition of ∆ ϑ (r) simplifies: in this case, all information about θ is captured by the sufficient statistic If we again let dt = 1/n, then we find that ∆(r) converges to a finite-dimensional Gaussian random vector with mean and covariance again, if the filters K i are modeled as independent draws from some fixed distribution, then the above normalized sums converge to their expectations, by the LLN.
Sentence 2: as in the intermediate-SNR regime, we see that inference can be dramatically simplified in this short-time setting.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As an example, consider the family of WFSTs T a , a 2 ⌃, with a single state Q = I = F = {0} and with T a defined by self-loop transitions with all input labels b 2 ⌃ with the same output label a, and with uniform weights.
Sentence 2: t a maps all labels to a.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this section, we will show that the function −D(•, •) is monotone and consistent.
Sentence 2: by Rizzi's result, it follows that we can find a 2-clustering {S 1 , S 2 } = {S 1 , S \ S 1 } that minimizes −D(S 1 , S 2 ), and hence maximizes D(S 1 , S 2 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is due to the term 1/(Kγ) in the bounds of Theorem 2 dominates both the bias and the MSE as expected since K is fixed.
Sentence 2: we observe a drop in the bias and the MSE as we increase γ up to ≈ 8 × 10 −5 , and then they gradually increase along with γ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We also consider the problem of maximizing the global gain from a set of initial states chosen according to some probability distribution P(x).
Sentence 2: accordingly, we define the global gain of a policy 11"": ]JL(7f) == Ex ]JL(x; 7f)P(x) and the global loss LIL of using some approximate policy 7r instead of the optimal one nIL (4) knowing the pdf over all parameters J-l we can define the expected global loss L == EJL [LIL].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To learn from unlabeled data, we observe that a Bayesian approach provides us with a predictive posterior distribution-i.e., we are able to quantify predictive uncertainty.
Sentence 2: we regularize the posterior by adding an unsupervised loss term that minimizes the predictive variance at unlabeled data points: where n and m are the numbers of labeled and unlabeled training examples, α is a hyperparameter controlling the trade-off between supervised and unsupervised components, and θ represents the model parameters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We define the max-margin majority voting (M 3 V) as solving the constrained optimization problem to estimate the true labels y and weights η: And in practice, the worker labels are often linearly inseparable by a single hyperplane.
Sentence 2: we relax the hard constraints by introducing non-negative slack variables {ξ i } M i=1 , one for each task, and define the soft-margin max-margin majority voting as inf ξi≥0,η,y , where c is a positive regularization parameter and − ξ i is the soft-margin for task t i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The high speed of human sensory perception [1] is puzzling given its inherent computational complexity: sensory inputs are noisy and ambiguous, and therefore do not uniquely determine the state of the environment for the observer, which makes perception akin to a statistical inference problem.
Sentence 2: the brain must represent and compute with complex and often high-dimensional probability distributions over relevant environmental variables.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition to having a more adaptive representation, the main benefit of SVDGP is that the computation of an unbiased gradient requires only linear complexity in M α , as shown below (see Appendix Bfor details).
Sentence 2: the time complexity to compute ∇ a KL[q||p] can be reduced to O(N m M α ) if we sample over the columns of K α with a mini-batch of size N m . By contrast, the time complexity to compute ∇ B KL[q||p] will always be Θ(M 3 β ) and cannot be further reduced, regardless of the parametrization of B.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It has been observed that in face recognition the variations between the images of the same face due to illumination and viewing direction are almost always larger than image variations due to the changes in face identity [1].
Sentence 2: while the PCA projections are optimal in a correlation sense (or for reconstruction" from a low dimensional subspace), these eigenvectors or bases may be suboptimal from the classification viewpoint.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Function ConsistentFinder(•): Let D st be the set of all directed paths from variable X s to variable X t . For any directed path d 2 D st , we define the weight of d as w d := ⇧ (u,v)2d b vu where b vu are coefficients in (1).
Sentence 2: by this definition, it can be seen that the entry (t, the entries of matrix A are multivariate polynomials of entries of B.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 1), we deduce that if the d-th feature of x is not to be used, it is sufficient and necessary for the d-th diagonal element of M be zero.
Sentence 2: analogous to the use of 1 -norm by the popular LASSO technique [34], we add the 1norm of M 's diagonal elements to the large margin metric learning criterion (M ) in eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then w 1 and w 2 are the i th columns of U and V respectively, and v T 1 v 2 is the i th diagonal element of S if v 1 and v 2 are the i th principal vectors of S 1 and S 2 .
Sentence 2: geometrically, this theorem says that after projection on the plane (P ) defined by any one of the principal vector pairs between subspaces S 1 and S 2 , both the entire subspaces collapse to just two lines such that points from S 1 lie along one line while points from S 2 lie along the second line.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that four of these examples are sufficiently hard to be unsolved by a variable elimination-based exact solver, even with several orders of magnitude more computational resources (200GB memory, 24 hour time limit).
Sentence 2: dIS provides excellent anytime behavior; in particular, (N l =1, N d =10) seems to work well, perhaps because expanding the search tree is slightly faster than drawing a sample (since the tree depth is less than the number of variables).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Some examples of these types of data are functional magnetic resonance imaging (fMRI) (Grosenick et al., 2008) and near-infrared spectra (NIR) (Xiaobo et al., 2010).
Sentence 2: it is important to develop methods for automatic induction that take into account the functional structure of the data (infinite dimension, high redundancy, etc.)
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let the gradient (8) vanish, then the primal iterates x k+1 can be exactly determined by which requires inverting an n × n positive definite matrix I + A T A, or equivalently, inverting an m × m positive definite matrix I + AA T via the following Sherman-Morrison-Woodbury identity, ) One basic fact is that we only need to invert such a matrix once and then use this cached factorization in subsequent iterations.
Sentence 2: there are several cases for which the above factorization can be efficiently calculated: (i) Factorization has a closed-form expression.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To get around this issue, we observe that an approximate solution for TAS lies in the row space of W T and therefore, an approximately optimal solution for AS lies in the row space of W T . We then set up the following regression problem: min X AS − XW T 2 F . Note, this regression problem is still too big to be solved in sublinear time.
Sentence 2: we sketch it by sampling columns of AS according to the leverage scores of W T to set up a smaller regression problem.
Relationship: reasoning
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: the impact of estimation error on the exactly optimal policy is minimal.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When the reference frame for an image is ambiguous, one factor that influences the inferred reference frame is the orientation of other images it is grouped with, especially when those images are identifiable in any orientation.
Sentence 2: if we ask people to solve an arithmetic problem, where the operator × is paired with the numbers 5 aligned with the top-down axes of the page (Figure 2 (a)), they should respond 25, the result of multiplication.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 24) the decoding variance in the three input conditions, which are (because of the symmetry, only the results for the network 1 are shown) where Var(z 1 |c vi ), Var(z 1 |c ve ) and Var(z 1 |c b ) denote, respectively, the decoding errors when only the visual cue, only the vestibular cue and both cues are presented.
Sentence 2: it is straightforward to check that in the weak input limit, the coupled CANNs implements Bayesian inference perfectly (compare Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a given x, we assume that once it pays the cost to acquire a feature, its value can be efficiently cached; its subsequent use does not incur additional cost.
Sentence 2: the cost of utilizing a particular prediction model, denoted by c(f z , x), is computed as the sum of the acquisition cost of unique features required by f z . Oracle Gating: Consider a general gating likelihood function q(z|x) with z ∈ {0, 1}, that outputs the likelihood of sending the input x to a prediction model, f z .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Optimizing the original energy in Eq.
Sentence 2: ( 1) is equivalent to optimizing its logarithmic form log E s . To efficiently solve this difficult optimization problem, we can instead optimize the lower bound of log E s as a surrogate energy, by applying Jensen's inequality: With f s (z) = z −s , s > 0, we observe that E log becomes sE 0 = s i =j log( ŵi − ŵj −1 ), which is identical to the logarithmic hyperspherical energy E 0 up to a multiplicative factor s. minimizing E 0 can also be viewed as a relaxation of minimizing E s for s > 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: An important quantity in this analysis is the level of non-redundancy between the hidden variables h and g. Let If Figure 1(a) is the correct induced topology among {z 1 , z 2 , z 3 , z 4 }, then the smaller ρ is, the greater the gap between ).
Sentence 2: ρ also governs how small the ∆ i,j need to be for the quartet test to return a correct pairing; this is quantified in Lemma 4.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Ceteris paribus, subjects will prefer a larger quantity of money to a smaller quantity of money.
Sentence 2: while the psychological de-sirability pointer could assign preferences to monetary labels capriciously (as an infant who prefers the drawings on a $1 bill to those on a $100 bill might), to model desirability behavior corresponding to knowledgeable use of currency, we constrain it to follow arithmetic ordering such that, where the notation x m denotes an item (currency token) x associated with the money label m. Then, Equation 3 reduces to, where max(M ) ≤ m * , since the contribution to p(r|x, m, c) for all larger m terms, is set to zero by the arithmetic ordering condition; the p(x|m) term binds x to all the m s it has been seen with before.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Micro-averaging: Micro-averaged multilabel performance metrics micro are defined by averaging over both labels and examples.
Sentence 2: let: c TN(f ) and c FN(f ) are defined similarly, then the micro-averaged multilabel performance metrics are given by: ) for micro-averaging, one applies a binary performance metric to the confusion matrix defined by the averaged quantities described in (5).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The filtered inputs to the two spiking models A and B are then N A * F L * S and N B * F R * S. These will be identical for any sound S whenever N A * F L = N B * F R , implying that the two neurons fire synchronously.
Sentence 2: for each location indicated by its filter pair (F L , F R ), we define the synchrony pattern as the set of binaural pairs of neurons (A, B) such that N A * F L = N B * F R . This pattern is location-specific and independent of the source signal S. the identity of the synchrony pattern induced by a binaural stimulus indicates the location of the source.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, complete coverage holds: We say there is an angular separation if cos(b i , b k ) ≤ 0 for any i, k = 1, . . . , K (i.e., the angles for all pairs are at least π/2), then ω ∈ 1 − r Rmax , 1 = ∅.
Sentence 2: under angular separation, the range ω that allows for full coverage is nonempty independently of K. Our result is in agreement with that of Nguyen (2015), whose result suggested that topic simplex B can be consistently estimated without knowing K, provided there is a minimum edge length a min > 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: But this process can be very costly, as the number of required comparison is proportional to 1/ 2 , even for strongly suboptimal arms.
Sentence 2: our algorithm, UnchainedBandits, when combined with decoys, first produces an "-approximation b P of P using a peeling approach and direct comparisons before refining it into P by using Algorithm 3 together with decoys.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, unlike other bandit problems, the argument inside the KL divergence is a product of parameters θ i κ l : While d KL (•, θ i κ l ) is convex to θ i κ l , it is not convex to the parameter space {θ i }, {κ l }.
Sentence 2: finding a set of parameters that minimizes i,l d KL (µ i,l , θ i κ l ) is non-convex, which makes PBMU difficult.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Throughout this pap er including exp eriments, we adopt the o rigi nal defi ni t ion of the F isher kernel from [6] .
Sentence 2: 3Notice t h at p-l (t) = log tlog(l -t ) by setting A Theoretical Analysis In this section , we compare the TOP kernel with the plugin es timate in terms of p erformance measures .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, recall the term which dominated our analysis of Q i under H 1 , ( ⇤ ) > ⌃ 2:d ⇤ , which was equal to We replace the covariances by their rescaled versions to obtain: For the spiked covariance model, rescaling variances to one amount to rescaling with D 1 = 1 /(1+✓).
Sentence 2: we see that our signal strength is affected only by constant factor (assuming ✓  1).
Relationship: reasoning