sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Suppose that we define the first order transition probabilities of a PARW by Then, we see that state i is an absorbing state (either partially or fully) when λ i > 0, and is a transient state when λ i = 0.
Sentence 2: the matrix Λ acts like a regularizer that controls the absorption rate of each state, i.e., the larger λ i , the larger p ii .
Relationship: entailment
Domain: NeurIPS
Sentence 1: At the first iteration all messages are initialised to constant functions.
Sentence 2: at each following iteration, each variable x i aggregates all incoming messages and computes the belief b i (x i ), which is then used to obtain the maxsum assignment x M S . for every variable The convergence of the max-sum is usually characterized considering fixed points for the message update rules, i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A GMRF is an undirected graphical probabilistic model that expresses a multi-dimensional joint Gaussian distribution in a reduced parameter space by making use of conditional independences.
Sentence 2: we employ a standard representation of a GMRF derived from the inverse covariance matrix, or precision matrix Q = Σ −1 of the underlying Gaussian distribution: For X = (X 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since S has an addition operation, we may parameterize h ∈ H explicitly as h a (x) = x + a for x ∈ u and parameter a ∈ v such that (u + a) ⊂ v. The restriction behavior of the translations in H prevents us from simply generating a group out of the elements of H. To get around this difficulty, we will decompose the h ∈ H into a composition of two functions: a translation group action and an inclusion.
Sentence 2: let S generate a group of translations T by defining the injective map to every element of a ∈ S we associate a member of the group T whose action corresponds to translation in S by a: A ta (x) = x + a for x, a ∈ S. (Although we assume the specific case of translations throughout, the sets of intrinsic operations H i may more generally contain other kinds of transformations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Prequential codes depend on the performance of the underlying training algorithm, and take advantage of the model's generalization ability from the previous data to the next.
Sentence 2: the model training should yield good generalization performance from data [1; In practice, optimization procedures for neural networks may be stochastic (initial values, dropout, data augmentation...), and Alice and Bob need to make all the same random actions in order to get the same final model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that given a distribution over an initial X 0 , we can apply an environment simulator multiple times to a distribution over the initial state, yielding a probability distribution over trajectories and cumulants.
Sentence 2: temporally abstract environment simulators only need to represent a relaxed version of the above conditional probability distribution: where τ is some arbitrary time skip interval up to the end of the trajectory, which can be chosen by the model and R τ t denotes the sum τ k=t R k . a temporally abstract environment simulator must only be able to predict some future state of the system and additionally provide the sum of the cumulants since the last step.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On line 4 we construct the collection of sets P that are the maximal sets that any vertex can take as parent set during the upper bound computation.
Sentence 2: the sets in P take immoralities relaxedly into consideration: For any v i , v j ∈ V , we have {v i , v j } ⊆ P for some P ∈ P if and only if when choosing parent sets during the upper bound computation, we allow immoralities to appear, as long as they are not between vertices of the fixed partial solution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The VCG mechanism, which we describe below, is the canonical strategy-proof mechanism, which means that every bidder's dominant strategy is to bid truthfully.
Sentence 2: for every Bidder i, no matter the bids made by the other bidders, Bidder i maximizes her expected utility (her value for her allocation minus the price she pays) by bidding her true value.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work we show that for many graphical models it is possible to devise a partitioning strategy followed by a message passing algorithm such that efficiency can be improved significantly.
Sentence 2: our approach adds additional terms to the energy function (i.e., regions to the Hasse diagram) such that the high-order factors can be pre-computed and remain constant during local message passing within each machine.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This equation is easily expressed in matrix form as Y = XH, where time is now indexed by matrix row in Y and X. H contains the concatenation of the terms and row t of X is similarly The standard solution for this regression problem is H = (X X) −1 X Y .
Sentence 2: the parameters of the kernels are derived using the regression technique by correlating the input and the output, and are compensated by the covariance in the input, i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We call the parameter α(k) ∈ (0, 1] the multitasking capacity of G for matchings of size k. Definition 1.1 generalizes to networks of depth r > 2, where instead of matchings, we consider first layer to last layer node disjoint paths, and instead of induced matchings we consider induced paths, i.e., a set of disjoint paths such that no two nodes belonging to different paths are adjacent.
Sentence 2: the main question we shall consider here is what kind of tradeoffs one should expect between α, d and k. which network architectures give rise to good multitasking behavior?
Relationship: entailment
Domain: NeurIPS
Sentence 1: To illustrate the complexities that can arise in this inferential setting, we adopt a latent variable network model with a standard flavor: a logistic regression model that simultaneously incorporates aspects of blockmodels, additive effects, and explanatory variables (see [10] for a more general formulation).
Sentence 2: we incorporate a K-class stochastic blockmodel component parameterized in terms of a symmetric matrix Θ ∈ R K×K and a membership vector z ∈ {1, . . . , K} N whose values denote the class of each node, with P ij depending on Θ zizj .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then we approximate the MAP classifier by using training data as a proxy for the latent sources.
Sentence 2: we take ratio (6), replace the inner sum by a minimum in the exponent, replace V + and V by R + and R , and replace D + by D to obtain the ratio: Plugging ⇤ (T ) in place of MAP in classification rule (5) yields the weighted majority voting rule (2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the model defined in the previous section, each branch (k → l) ∈ T has a collection of contextdependent rule probabilities θ k→l .
Sentence 2: θ k→l specifies a collection of multinomial distributions, one for each C = (c l , x, c r ), where c l is left phoneme, x is the old phoneme, c r is the right phoneme.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we study the problem under the assumption that the predictors have a linear form and the loss functions are quadratic.
Sentence 2: f ⇢ Lin(X , W) where W = {w 2 R d | kwk ⇤  w lim } and X = {x 2 R d | kxk  x lim } for some given constants w lim , x lim > 0, while `t(y) = (y y t ) 2 , where |y t |  x lim w lim .
Relationship: entailment
Domain: NeurIPS
Sentence 1: To simplify notations in our description, we will replace the optimization problem in (1) with a more general formulation.
Sentence 2: instead of working with n input data vectors x i ∈ R d , we work with d feature vectors f j ∈ R n (j = 1, . . . , d), and y ∈ R n . Each f j corresponds to the j-th feature component of x i for i = 1, . . . , n. f j,i = x i,j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: If i) e f λ is as in Corollary 2, i.e., e f λ (||x|| 2 ) = g(min{||x|| 2 , λ}), ii) X n has a (ρ 1 , ρ 2 )-balanced subset of samples I with k balls, and iii) the centers of the balls are at least v > 4r + 2g −1 ( ρ1+ρ2 ρ1 g(r)) apart, then for λ ∈ the set of outliers X n \I has no effect on the grouping of inliers I.
Sentence 2: if {x, y} ∈ B l and {c 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, on all datasets, the treeRMNs with CVI can achieve similar prediction quality as the existing methods (i.e., MSL, IRM and MRC), but is about two orders of magnitude more efficient in training.
Sentence 2: it achieves significant improvements on the Animal and Nation data, but moderately worse results on the UML and Kinship data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, the spatial map M 1 : Ω 0 → R p1 introduced above can be obtained by (i) computing the quantities Z x for all patches x of the image I (spatial convolution after mirroring the filters z j ); (ii) contrast-normalization involving the norm x ; (iii) applying the pointwise non-linear function κ 1 ; (iv) applying the linear transform κ 1 (Z Z) −1/2 at every pixel location (which may be seen as 1×1 spatial convolution); (v) multiplying by the norm x making ψ 1 homogeneous.
Sentence 2: we obtain a particular convolutional neural network, with non-standard parametrization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: No matter the model satisfies the assumption or not, an easy way to fulfill it is to add an additional term to l(p), where µ is a small positive number.
Sentence 2: for each s, we make an I i = {1, . . . , k} with I + i = {s}, r i = µ, and r ′ i = 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The specific and well-studied prediction market mechanism we consider is the potential-based market maker [1].
Sentence 2: here, traders interact with the market maker sequentially, and the cost for each trade is determined by a convex potential function C : R → R applied to the market maker's state s ∈ R. the cost for a trade dr when the market maker has state s is given by cost(dr; s) = C(s−dr)−C(s), i.e., the change in potential value of the market maker's position due to the market maker accepting the trade.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Despite each application domain has its own specificity, 4 in this paper we abstract such details away, and focus on the broad problem of building matches between the two parties in the reciprocal recommendation problem based on behavioral information only.
Sentence 2: we do not consider explicit user preferences (e.g., those evinced by user profiles), but only the implicit ones, i.e., those derived from past user behavior.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is obtained by adding 1 if the new breakpoint is before J t−1 and 0 otherwise.
Sentence 2: if there is no minimum of P t (µ) in piece p t,jt , we must move the pointer from j t to its final position J t ∈ {1, ..., t + 1}, where J t is the index of the rightmost function piece that contains a minimum: See Figure 2 for an example.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One approach to deal with this issue has been to investigate special cases where these problems would be tractable.
Sentence 2: the basic idea is to select models from a restricted class of Bayesian networks that have structural properties enabling fast learning or inference; this way, the computational complexity will not be an issue, though possibly at the cost of accuracy if the true distribution is far from the model family.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In other words, the timevarying coverage function controls the propensity of occurring events over time.
Sentence 2: for a fixed set S i , as time t increases, the cumulative number of events observed grows accordingly for that f (S i , t) is a continuous monotonic function over time; for a given time t, as the set S i changes to another set S j , the amount of coverage over domain U may change and hence can result in a different cumulative intensity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, if Y := {−1, 1} and L is a convex, margin-based loss represented by ϕ : R → [0, ∞), that is L(y, t) = ϕ(yt) for all y ∈ Y and t ∈ R, then L can be clipped, if and only if ϕ has a global minimum, see [13,Lemma 2.23].
Sentence 2: the hinge loss, the least squares loss for classification, and the squared hinge loss can be clipped, but the logistic loss for classification and the AdaBoost loss cannot be clipped.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In contrast to most existing trackers which only learn the appearance of the tracked object online, we take a different approach, inspired by recent advances in deep learning architectures, by putting more emphasis on the (unsupervised) feature learning problem.
Sentence 2: by using auxiliary natural images, we train a stacked denoising autoencoder offline to learn generic image features that are more robust against variations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: CNN-based methods [2][3][4] show strong cross-scene prediction capability, due to the learning capacity of CNNs.
Sentence 2: [3] uses a multi-column CNN with different receptive field sizes in order to encourage different columns to capture features at different scales (without input scaling or explicit supervision), while [4] uses a pyramid of input patches, each sent to separate sub-network, to consider multiple scales.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From the above Q dev p values we can compute player p's value for deviating from an equilibrium which recommends action a in state s: it is Q dev p (s, a ) for the best possible deviation a , since p will get the one-step payoff for a but be punished by the rest of the players starting on the following time step.
Sentence 2: v dev p (s, a) is the value we must achieve for player p in state s if we are planning to recommend action a and punish deviations with π pun p : if we do not achieve this value, player p would rather deviate and be punished.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In such a setting, we cannot use GREEDI as presented above, since we cannot evaluate f on the individual machines without access to the full set V . Fortunately, many such functions have a simple structure which we call decomposable.
Sentence 2: more precisely, we call a monotone submodular function f decomposable if it can be written as a sum of (non-negative) monotone submodular functions as follows: there is separate monotone submodular function associated with every data point i ∈ V . We require that each f i can be evaluated without access to the full set V . Note that the exemplar based clustering application we discussed in Section 3.1 is an instance of this framework, among many others.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the deterministic rule, i k is an arbitrary sequence that is assumed to be essentially cyclic.
Sentence 2: there is an N ∈ N, N ≥ N , such that each block i ∈ {1, 2, . . . , N } is updated at least once in a window of N , that is, This encompasses different kinds of cyclic rules such as fixed ordering, random permutation, and greedy selection.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In contrast, experimental and theoretical studies of neural systems suggest memory retrieval is a dynamic and iterative process: memories are retrieved through a potentially varying period of time, rather than a single step, during which information can be continuously integrated [3,7,20].
Sentence 2: attractor dynamics are hypothesised to support the robust performance of various forms of memory via their self-stabilising property [8,12,16,28,33].
Relationship: entailment
Domain: NeurIPS
Sentence 1: (2) If the Gram matrix is entry-wise non-negative and if the set of predictors indexed by {1, . . . , p} can be partitioned into subsets B 1 , .
Sentence 2: . . , B B such that this applies to design matrices whose entries X ij = φ j (u i ) contain the function evaluations of non-negative functions {φ j } p j=1 traditionally used for data smoothing such as splines, Gaussians and related 'localized' functions at points {u i } n i=1 in some fixed interval, see Figure 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Example 1 (Plackett-Luce as an RUM [2]) In the RUM, let µ j 's be Gumbel distributions.
Sentence 2: for alternative j ∈ {1, . . . , m} we have µ j (x j |θ j ) = e −(xj −θj ) e −e −(x j −θ j ) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We then apply softmax normalization to the computed weights so that each element of the context is computed as a weighted average over center elements, rather than just a weighted sum.
Sentence 2: r(Φ, φ) (j) = e (W φ+b) (j) / |Φ| k=1 e (W φ+b) (jk) Φ with weights W ∈ R |φ|×|Φ|×|c| , and biases b ∈ R |Φ|×|c| .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We call this algorithm OOMM (Oblivious Online Match Maker), The main idea is to develop a strategy that is able to draw uniformly at random as many pairs of reciprocal edges as possible from E r (recall that E r is the set of all reciprocal edges between B and G).
Sentence 2: within the T rounds, OOMM will draw uniformly at random Θ(T )-many such pairs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use a simplicial complex K to model such a space.
Sentence 2: given input sampled points P and the distance matrix of P , we first construct a proximity graph based on either r-neighborhood or k-nearest neighbors(NN) information; that is, a point p ∈ P is connected either to all its neighbors within r distance to p, or to its k-NNs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, in a high-dimensional setting, where the number of covariates p could scale with the number of samples n, this performance of the LDA classifier breaks down.
Sentence 2: bickel and Levina [3] show that when p/n → ∞, then the LDA classifier could converge to an error rate of 0.5, that of random chance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let f ′′ t denote the resulting hypothesis, that is, Setting σ i,t+1 = φ t σ i,t for all i ∈ I ′ t , we can write, The third and last step of the update is the removal step discussed above.
Sentence 2: if the budget constraint is violated and |I ′ t | > B then I t+1 is set to be I ′ t \ {r t } where r t = min I ′ t . Otherwise, I t+1 simply equals I ′ t . The recursive definition of the weight σ i,t can be unraveled to give the following explicit form, σ i,t = j∈It−1 ∧ j≥i φ j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use the notation nn for 'neural network' to indicate that this is a nAD.
Sentence 2: m q is the identifier of a neural network model that specifies a probability distribution over its output values u, given input t. from the perspective of the probabilistic logic program, an nAD realizes a regular AD p 1 :: q( t, u 1 ); ...; p n :: q( t, u n ) :− b 1 , ..., b m , and DeepProbLog thus directly inherits its semantics, and to a large extent also its inference, from ProbLog.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , X p ) 2 X p ✓ R P . Each coordinate of the random vector X is associated with a vertex in V and the graph structure encodes the conditional independence assumptions underlying the distribution of X.
Sentence 2: x a and X b are conditionally independent given all the other variables if and only if (a, b) 6 2 E, that is, the nodes a and b are not adjacent in G. One of the fundamental problems in statistics is that of learning the structure of G from i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a general hypothesis testing problem, namely, H 0 : θ ∈ G 0 versus H 1 : θ ∈ G 1 , the minimax testing risk with respect to an algorithm A and a statistical oracle r ∈ R[ξ, n, T n , η(Q A )] is defined as  • For any sequence {γ n } n≥1 satisfying γ n = Ω(γ * n ), there exist a constant η > 0 and an algorithm Throughout this paper, we assume that the covariance matrix Σ in (2.1) is known.
Sentence 2: for some positive definite Σ ∈ R d×d , the parameter spaces of the null and alternative hypotheses are defined as Accordingly, the testing problem of detecting whether µ 0 � = µ 1 is to distinguish In §3.1, we present the minimax rate of the detection problem from an information-theoretic perspective.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We consider learning a weight matrix consisting of the weight vectors for m linear predictive models: In this paper, we propose a non-convex multi-task feature learning formulation to learn these m models simultaneously, based on the capped-ℓ 1 ,ℓ 1 regularization.
Sentence 2: we first impose the ℓ 1 penalty on each row of W , obtaining a column vector.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition to providing fast general feature extraction for 360°imagery, our approach provides a bridge from 360°content to existing heavily supervised datasets dedicated to perspective images.
Sentence 2: training requires no new annotations-only the target CNN model (e.g., VGG [33] pre-trained on millions of labeled images) and an arbitrary collection of unlabeled 360°images.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Structural risk minimization schemes over LR via various forms of regularizations have been widely studied and understood in the literature.
Sentence 2: recent asymptotic analysis of the 1 norm and 1 / 2 mixed norm regularized LR proved that under certain conditions the estimated sparse coefficient vector β enjoys a property called sparsistency [34], suggesting their applicability for meaningful variable selection in high-dimensional feature space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Model test (fr-en) test (de-en) Bilingual Enc-Dec [Bahdanau et al., 2015] 35.2 27.3 Multi-lingual Enc-Dec [Johnson et al., 2016] 35 As shown in the third row of Figure 2, on German and Adult, we achieve higher accuracy on the biased categories, even though our overall accuracy is similar to or lower than the baseline which does not employ fairness constraints.
Sentence 2: on Adult, our performance on the biased categories is 0.788 while the baseline's accuracy is 0.748.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the CoffeeScript-JavaScript task, we create several datasets named as XY-ZW: X and Y (C or J) indicate the source and target languages respectively; Z (A or B) indicates the vocabulary; and W (S or L) indicates the program length.
Sentence 2: vocabulary A uses {x,y} as variable names and {0,1} as literals; vocabulary B uses all alphabetical characters as variable names, and all single digits as literals.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, in WNNM [15], j∈Ci Φ(X) j i G(X) j corresponds to the projection of X i onto the group-specific subspace as a function of the selected patches.
Sentence 2: the subspace for calculating Z i is spanned by the eigenvectors , where diag{σ} is obtained by applying the shrinkage function associated with the weighted nuclear norm [15] to the eigenvalues of X T Ci X Ci .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The noise precision θ 2 does not change with N . Identical to [9], the transform A is modeled as a large, right-orthogonally invariant random matrix.
Sentence 2: we assume that it has an SVD of the form (9) where U and V are N × N orthogonal matrices such that U is deterministic and V is Haar distributed (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: LSI uses singular value decomposition to discover the important associative relationships of word features [10], and create a reduced-dimension feature space.
Sentence 2: we first perform singular value decomposition over M * , M * = U SV ⊤ , and then obtain a low dimensional representation matrix Z via a projection Z = M * V k , where V k contains the top k right singular vectors of M * .
Relationship: entailment