sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: A quantum state vector is a complex unit vector whose components have magnitudes that are equal to the square-roots of the probabilities.
Sentence 2: if ψ is a quantum state vector and p is the corresponding probability distribution then where the unit column vector ψ is called the quantum state vector which sits in the vector space C n , • is the Hadamard (pointwise) product and † is the complex conjugate transpose.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider a map in the form of an undirected graph G and let X be a set of binary variables, which are in one-to-one correspondence with the edges of graph G. For example, the graph in Figure 2 will lead to 12 binary variables, one for each edge in the graph.
Sentence 2: a variable instantiation x will then be interpreted as a set of edges in graph G. instantiation x includes edge e iff the edge variable is set to true in instantiation x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although seemingly unrelated at first glance, the layers of a deep neural network (DNN) can be viewed as iterations of some algorithm that have been unfolded into a network structure [9,11].
Sentence 2: iterative thresholding approaches such as IHT mentioned above typically involve an update rule comprised of a fixed, linear filter followed by a non-linear activation function that promotes sparsity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Learning structure and global mixtures: To simultaneously learn global mixture labels m n and edge structures associated with each mixture E m , we use an EM algorithm for learning mixtures of trees [20,15].
Sentence 2: meila and Jordan [20] describe an EM algorithm that iterates between inferring distributions over tree mixture assignments (the E-step) and estimating the tree structure (the M-step).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To make the bandit version of FTAL -differentially private, we replace the value v t = t τ =1 p β f t (w † t + βu t )u t with a private approximation v † t computed using the tree-based sum protocol.
Sentence 2: at each time step t we output See Algorithm 3 (Appendix E.1) for details.
Relationship: entailment
Domain: NeurIPS
Sentence 1: from K k=1 w zmk δ k , and therefore there is nothing in the model that encourages the image features, x ml and x ml ′ , which are associated with the same image-feature atom θ * j , to be assigned to the same object k. To address this limitation, we add a clustering step within each of the images; this is similar to the structure of the hierarchical Dirichlet process (HDP) [21].
Sentence 2: consider the following augmented model: where v m ∼ Stick T (α v ), and G k is as defined in (3).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This suggests an interesting hypothesis why choosing a binary code with very high temporal precision might be advantageous even if the signal of interest by itself does not change at that time scale: the sensory inputmay sometimes be too noisy, so that repeated, independent samples from the signal of interest may sometimes lead to neuronal firing and sometimes not.
Sentence 2: a binary code at the short time scale is useful independent from the correlation time of the signal to be encoded, if uncertainties have to be taken into account, because any surplus available amount of temporal precision is maximally used for uncertainty representation in a self-adjusting manner.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, suppose that after each iteration t, each player i observes a partial payoff function ũi t : S i → R describing their payoff as a function of only their own action, s i , holding the action played by the other player fixed.
Sentence 2: ũ1 t (s 1 ) := u(s 1 , s 2 t ) and ũ2 t (s 2 ) := −u(s 1 t , s 2 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Even with a small, sampled, set of template orbits, not including the actual orbit of x, we can still compute an invariant signature.
Sentence 2: observe that when g is unitary gx, τ k = x, g −1 τ k . the inner product of the transformed image with a template is the same as the inner product of the image with a transformed template.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The infinite Tucker decomposition (InfTucker) generalizes the Tucker model to infinite feature space via a tensor-variate Gaussian process (TGP) [19].
Sentence 2: in a probabilistic framework, we assign a standard normal prior over each element of the core tensor W, and then marginalize out W to obtain the probability of the tensor given the latent factors: p(M|U (1) , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In a recent paper, Chambolle and Pock [11] proposed a primal-dual algorithm (PDA) for the following convex-concave saddle-point problem: ρ and h are possibly non-smooth convex functions, f is a convex function and has a Lipschitz continuous gradient defined on dom ρ with constant L, and T is a linear map.
Sentence 2: briefly, given x 0 , y 0 and algorithm parameters ν x , ν y > 0, PDA consists of two proximal-gradient steps: where D x and D y are Bregman distance functions corresponding to some continuously differentiable strongly convex ψ x and ψ y such that dom ψ x ⊃ dom ρ and dom ψ y ⊃ dom h. and D y is defined similarly.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, these works exclusively focused on using coding at the computation level, i.e., certain linear computational steps are performed in a coded manner, and explicit encoding/decoding operations are performed at each step.
Sentence 2: [13] used MDS-coded distributed matrix multiplication and [7] focused on breaking up large dot products into shorter dot products, and perform redundant copies of the short dot products to provide resilience against stragglers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The trace norm 1 X Σ is the sum of the singular values of X.
Sentence 2: the characterization in terms of the singular value decomposition allows us to characterize low trace norm matrices as the convex hull of bounded-norm rank-one matrices: the trace norm is a convex function, and the set of bounded trace norm matrices is a convex set.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, we adopt a different approach, which is based on a conjugate gradient method.
Sentence 2: we consider an approximate time-varying linear process with the same diffusion term, that is the same system noise covariance: where g(t, x) = −A(t)x+b(t), with A(t) ∈ R d×d and b(t) ∈ R d . the approximate posterior process q(X|Σ) is restricted to be a Gaussian process [4].
Relationship: entailment
Domain: NeurIPS
Sentence 1: By explicitly modeling and learning the output structure and task structure, our model also addresses the limitations of the existing models that typically assume certain specific type of output structures (e.g., tree [13]) or task structures (e.g., shared sparsity [11]).
Sentence 2: a model with task relatedness structure based on shared sparsity on the task weight vectors may not be appropriate in many real applications where all the features are important for prediction and the true task structure is at a more higher level (e.g., weight vectors for some tasks are closer to each other compared to others).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, we show that it can be written as Rg, where rows of R are "near-orthogonal" (with high probability) and have norm ||z|| 2 , and the vector g is close to Gaussian (g has independent sub-Gaussian elements), and hence the projections behave "near-independently".
Sentence 2: g = vec(D 1 ) (vector of diagonal entries of D 1 ), and R is a function of D 2 , D 3 and z. Theorem 4.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our second result demonstrates the deviation between the empirical means of Pr ′ and β(x) Pr in feature space, given β(x) is chosen perfectly in the population sense.
Sentence 2: this result shows that convergence of these two means will be slow if there is a large difference in the probability mass of Pr ′ and Pr (and thus the bound B on the ratio of probability masses is large).
Relationship: entailment
Domain: NeurIPS
Sentence 1: When learning a function f , we may have access to not only the output values f (x i ) for training points x i , but also the values of its j-th order derivatives with respect to the input, D j x f (x i ).
Sentence 2: instead of the typical training set consisting of pairs {(x i , f (x i ))} N i=1 we have access to In this situation, the derivative information can easily be incorporated into training a neural network model of f by making derivatives of the neural network match the ones given by f . Considering a neural network model m parameterised with θ, one typically seeks to minimise the empirical error in relation to f according to some loss function When learning in Sobolev spaces, this is replaced with: where j are loss functions measuring error on j-th order derivatives.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In section 3.2, we improve upon [4], by showing that (in this generalized framework) the optimal iteration complexity of oblivious, deterministic or stochastic, finite sum algorithms with both first-order and coordinate-descent oracles cannot perform better than Ω(n + κ ln(1/ )), unless the strong convexity parameter is provided explicitly.
Sentence 2: the richer expressiveness power of this framework allows addressing incremental gradient methods, such as Incremental Gradient Descent [6] and Incremental Aggregated Gradient [8,IAG].
Relationship: entailment
Domain: NeurIPS
Sentence 1: These algorithms differ from the standard value function based ADP methods in how the greedy policy is computed.
Sentence 2: at each iteration CbPI algorithms approximate the entire greedy policy as the output of a classifier, while in the standard methods, at every given state, the required action from the greedy policy is individually calculated based on the approximation of the value function of the current policy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: After performing such inference, we retain those columns of Ψ that are used in the representation of the data in D, thereby inferring D and hence M . To impose our desire that the vector of dictionary weights α is sparse, one may adjust the parameters a and b.
Sentence 2: as discussed in [17], in the limit K → ∞, the number of elements of z i that are non-zero is a random variable drawn from Poisson(a/b).
Relationship: entailment
Domain: NeurIPS
Sentence 1: All these problems of linear averaging can be well addressed by a KL-averaging method which averages the model (instead of the parameters) by finding a geometric center of the local models in terms of KL divergence (Merugu and Ghosh, 2003;Liu and Ihler, 2014).
Sentence 2: it finds a model Liu and Ihler (2014) studied the theoretical properties of the KL-averaging method, and showed that it exactly recovers the global MLE, that is, θ * KL = θmle , when the distribution family is a full exponential family, and achieves an optimal asymptotic error rate (up to the second order) among all the possible combination methods of { θk }.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this case, assuming that Θ = R d , the algorithm will reduce to those described in the papers [8,12].
Sentence 2: the maximum quasi-likelihood estimator becomes the least-squares estimator and as noted earlier, the algorithm behaves identically to one which chooses the parameter optimistically within the confidence ellipsoid {θ : θ − θt Mt ≤ ρ(t)}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: s nd = s nd ′ for some d = d ′ .
Sentence 2: the proposed model is an infinite mixture model, where the probability for the dth view of the nth instance is given by where θ n = {θ nj } ∞ j=1 are the mixture weights, θ nj represents the probability of choosing the jth latent vector, α is a precision parameter, N (µ, Σ) denotes the Gaussian distribution with mean µ and covariance matrix Σ, and I is the identity matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the context of search engines, it has been argued (e.g., by [12,18,1]) that a user's clicking behavior provides implicit feedback of a specific form on the ranking.
Sentence 2: since users will typically read the search results from first to last, when a user skips some links that appear earlier in the ranking, and instead clicks on a link that appears later, her action suggests that the later link was more informative or relevant.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Varying λ allows us to 'interpolate' between the two extremes of minimum dictionary cost and minimum pointer cost.
Sentence 2: λ can be interpreted as tracing out a regularization path that allows a more flexible representation of D. To formalize our compression criterion, let S = { x i . . . be the set of all unique k-grams in D, and P = { (s, l) | s = x l . . . x l+|s|−1 } be the set of all m = |P| (potential) pointers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Algorithm 2: CLAP: Confidence Lift And Project  We will assume that R = (R 1 , .
Sentence 2: . . , R m ) is a covering of G (in the sense introduced in the previous section), and -without loss of generality-we will assume that the edge set is E is the maximal set of edges that is compatible with R being a covering.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The simple potential function is defined as where λ > 0 is a hyperparameter that controls the magnitude of reward shaping.
Sentence 2: when the state s is not the terminal state S ⊥ , ϕ(s) carries a scaled version of the number of positive symptoms that have been identified before entering state s. By the definition f (s, s ) := γϕ(s ) − ϕ(s), scaling up λ would be equivalent to scaling up f to make the auxiliary rewards larger in magnitude.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The value of a policy under rewards It is expressed as the weights multiplied by the accumulated discounted feature value given a policy, which we define as MDP solver We assume that there is a RL algorithm ALG that takes an MDP as input and outputs an optimal policy and its corresponding representation in the feature space.
Sentence 2: aLG(M ) Maxmin Learning All weights that are consistent with the agent's knowledge form a set P R . We will discuss several formal ways to define this set in Section 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To further validate the consistency of layout distribution between the ground truth and inserted instances by our method, we evaluate whether RGB images rendered using the inserted masks can be reasonably recognized by a state-of-the-art image recognizer, e.g., YOLOv3 [21].
Sentence 2: given an RGB image generated by [31], we check whether the YOLOv3 detector is able to detect the inserted instance in the rendered image as shown in Figure 8 and Figure 9.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The surrogate likelihood (which is convex), was used to initialize the univariate and clique likelihood, while the univariate likelihood was used to initialize the smooth classification error.
Sentence 2: 1 There are two redundancies, as adding a constant to a(yi, yj ) or c(yi) has no effect on p. 2 dL da(y,y ′ ) = P (i,j) dL dθ(y i =y,y j =y ′ ) , dL db(y) = P i dL dθ(y i =y) xi, and dL dc(y) = P i dL dθ(y i =y) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Figure 4(b) shows the effect that the feature dimensionality (controlled by the codebook size of OMP-1) has on the performance of the linear and non-linear classifiers, and Table 1 provides representative numerical results.
Sentence 2: when the codebook size is low, the assumption that we can approximate the non-linear function f as a globally linear classifier fails, and in those cases the R 2 SVM and RBF SVM clearly outperform the linear SVM.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We then define the connecting module as a classifier, which takes the left and right phrases P (l) and P (r) as input and outputs a normalized score S(P (m) j | P (l) , P (r) , I) for each j ∈ {1, . . . , L}.
Sentence 2: we adopt a two-level LSTM model [2] to encode P (l) and P (r) respectively, as shown in Figure 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The main idea is to use the Markov property to eliminate the dependence of the confidence intervals on the unknown quantities (including π ⋆ and γ ⋆ ).
Sentence 2: we estimate the transition probabilities from the sample path using simple frequency estimates: as a consequence of the Markov property, for each state, the frequency estimates converge at a rate that depends only on the number of visits to the state, and in particular the rate (given the visit count of the state) is independent of the mixing time of the chain.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In other words, Kernel PCA is a generalization of conventional PCA since different kernels can be utilized for different nonlinear projections.
Sentence 2: we can now project the vectors in Rf to a lower dimensional space spanned by the eigenvectors weI>, Let x be a test sample whose projection is <t>(x) in Rf, then the projection of <t>(x) onto the eigenvectors weI> is the nonlinear principal components corresponding to <t>: we can extract the first q (1 ~q ~m) nonlinear principal components (Le., eigenvectors w4» using the kernel function without the expensive operation that explicitly projects the samples to a high dimensional space Rf" The first q components correspond to the first q non-increasing eigenvalues of (6).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We represent conditional distributions in the RKHS using conditional mean embeddings [18].
Sentence 2: given observations {(x i , y i )} n i=1 of a pair of random variables (X, Y ), we construct the embeddings of the two sets of conditional distributions, {p(X|Y = y i )} n i=1 and {p(Y |X = x i )} n i=1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The gap between the log-likelihood and the ELBO is the Kullback-Leibler (KL) divergence between q(z) and the posterior p θ (z|x): log p θ (x) − L(q, θ; x) = D KL (q(z)||p θ (z|x)).
Sentence 2: (6) the amortized variational inference approach introduces a conditional variational distribution, z ∼ q φ (•|x), to approximate the intractable posterior distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This intriguing function of primate brain inspires us to develop a novel deep neural net, called multi-view perceptron (MVP), which can disentangle identity and view representations, and also reconstruct images under multiple views.
Sentence 2: given a single face image of an identity under an arbitrary view, it can generate a sequence of output face images of the same identity, one at a time, under a full spectrum of viewpoints.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The final set of classifiers C contains stochastic classifiers, whose output only depends on the outputs of the view-specific classifiers.
Sentence 2: associated to a set of classifiers C, there is a function For simplicity, in the rest of the paper, when the context is clear, the function x → Φ C (h 1 , ..., h V , x) will be denoted by c h1,...,hV .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assuming a simple white Gaussian noise model, this can be achieved through the following loss While the supervised learning of analysis operator has been considered for solving denoising problems [5,17], here we address more general scenarios.
Sentence 2: we argue that, when used along with metric learning, it is often better suited for classification tasks than its synthesis counterpart, because the non-generative nature of analysis models is more suitable for feature learning.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider the effect on dX (¡ a of perturbing the affinity 0 1' ) for the X t a -edge, to 0 1' ) §¦ (' ) .
Sentence 2: we show in the Appendix that the derivative of X 1¡ fX ¦ 1' ) a wa with respect to ¦ (' ) , evaluated at ¦ 1' ) ¥ # , satisfies Here X w ) 5a are the X t a elements of eigenvector x and X 8 % %8 ) 5a are degrees of nodes X t a (Eq.1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, Proposition 2 in [19] states that that there exist constants κ l 1 and κ l 2 , independent of n, p such that with probability at least 1 − c 1 exp(−c 2 n), for some constants c 1 , c 2 > 0, if n > 4(κ l 2 /κ l 1 ) 2 ηd log(p).
Sentence 2: with probability at least 1 − c 1 exp(−c 2 n), the loss function L satisfies RSC(k) with parameter κ l 1 provided n > 4(κ l 2 /κ l 1 ) 2 ηd log(p).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given two graphs G and H, let and likewise H j .
Sentence 2: let K GG , K HH , and K GH be the kernel matrices, induced by the embedding kernel k. they are defined such that Let G be the set of graphs with attribute domains A 1 , A 2 , ..., A L . Let G and H be two graphs in G. Let m G and m H be the corresponding graph embeddings.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (1)(left) with and Here, the residual Jacobian portion for the parameters of the integrating unit was omitted because they were merely fine-tuned with a steepest-descent type method owing to our knowledge-based design for input-partition to avoid (too many) local experts.
Sentence 2: the spectral reflectance signals (16 inputs) were converted to the hue angle as input to the integrating unit that consists of five bell-shaped basis functions, partitioning that hue-subspace alone in a fuzzy fashion into only five color regions (red, yellow, green, blue, and violet) for five 16-16-10 MLP-experts, each of which receives all the 16 spectral signals as input [hence, Z=5; n=2,210 (CA=85; n B =1,360); mlu dn =6.5].
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a first step, we generalize the Kesler's Construction originally defined for single-label classification (see [6]) to the PLM setting, thus showing that the embedding problem can be formulated as a binary classification problem in a higher dimensional space when new variables are appropriately defined.
Sentence 2: consider the vector y(a) = (M ωs(a) − M ωe(a) ) ∈ R s defined for every preference arc in a given preference constraint, that is a = (ω s , ω e ) ∈ A(g).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Recurrent neural networks (RNNs) live at the heart of many sequence modeling problems.
Sentence 2: the incorporation of gated additive recurrent connections is extremely powerful, leading to the pervasive adoption of models such as Gated Recurrent Units (GRU)  or Long Short-Term Memory (LSTM) [Hochreiter and Schmidhuber, 1997] across many NLP applications Rocktäschel et al., 2015;McCann et al., 2017].
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, in this paper we will focus on a useful property of this type of factorization that seems to have passed unnoticed thus far.
Sentence 2: we call it the "stochastic-factorization trick": Given a stochastic factorization of a square matrix, P = DK, swapping the factors of the factorization yields another transition matrix P = KD, potentially much smaller than the original, which retains the basic topology and properties of P. The stochasticity of P follows immediately from the same property of D and K. What is perhaps more surprising is the fact that this matrix shares some fundamental characteristics with the original matrix P. it is possible to show that: (i) for each recurrent class in P there is a corresponding class in P with the same period and, given some simple assumptions about the factorization, (ii) P is irreducible if and only if P is irreducible and (iii) P is regular if and only if P is regular (see [9] for details).
Relationship: entailment