sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: It has been shown that the MMV model provably improves the standard CS recovery by exploiting the block-sparse structure [10,21].
Sentence 2: in the MMV model we consider the reconstruction of the signal represented by a matrix W ∈ IR d×n , which is given by a dictionary (or measurement matrix) A ∈ IR m×d and multiple measurement vector B ∈ IR m×n such that Each column of A is associated with an atom, and a set of atom is called a dictionary.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The RPFB is an alternative approach that we describe in Section 3.
Sentence 2: in the fixed-window history-based approach (or window-based, for short), we only look at a fixed window of the immediate past values of X 1:t . we use samples in the form of Z t X t−H+1:t with a finite integer H that determines the length of the window.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In [8], our setting, generalized to pre-partitions of any number of subsets, was addressed by a heuristic extension of the probabilistic IB algorithm, yielding improved empirical results.
Sentence 2: the algorithm in [8] was based on a modification of the IB stable-point equation, which amplified the impact of features characterizing a formed cluster across all, or most, subsets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As additional input, the network is provided with a task-specific embedding vector.
Sentence 2: we have f (x, t; θ µ ) = h [x; ψ t ] T , θ h , where h(•), again, denotes the output of the neural network (here with parameters θ h ) and ψ t is the t-th row of an embedding matrix ψ ∈ R K×L (we choose L = 5 for our experiments).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Otherwise, for instance, if , it is impossible to distinguish between the case when x * 3 = 1 and the case when x * 2 = λ 1 and x * 1 = λ 2 .
Sentence 2: we do not restrict the feature matrix to be non-negative, which is more general than the traditional NMF and is potentially useful for many applications.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This means that the unfavorable cases, i.e., the Brownian realizations for which M − M ε > ε, are not concentrated on some subsets of Brownian realizations matching some evaluations in t 1 , ..., t Nε .
Sentence 2: the PAC guarantee also holds when restricted to the Brownian realizations matching the evaluations in t 1 , ..., t Nε only.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Computing a basis for the nullspace of the coefficient matrix in formulation (5) of the HMM filter recovers, according to Lemmas 1 and 2, the direction of one column of B.
Sentence 2: the direction of the column corresponding to observation y k , i.e., b y k . From such basis vectors, we can construct a matrix C ∈ R X×Y where the yth column is aligned with b y .
Relationship: entailment
Domain: NeurIPS
Sentence 1: As shown in Proposition 4.2, the first term in (4.4) is the quadratic form associated with the population Hessian matrix evaluated at Θ.
Sentence 2: such a quadratic form is universally lower bounded by the second term, which does not depend on Θ, and is further lower bounded by β/(d under the conditions of Proposition 4.2, the population loss function defined in (4.3) is strongly convex.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We define the set of all weights that satisfy the -soft-margin condition with respect to a distribution P via: Of particular interest is W( , Dn ), the set of all the weights that satisfy the -soft-margin condition with respect to the empirical data distribution.
Sentence 2: any w ∈ W( , Dn ) predicts with at least a margin on all but a ( ) fraction of the data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition to the transition probabilities, we now assume that the probability distribution of cost has uncertainty.
Sentence 2: for each pair of s ∈ S n and a ∈ A(s), the true probability mass function 2 , f (•|s, a), for the random cost, C(s, a), is in the following uncertainty set that is characterized with the nominal probability mass function, f 0 (•|s, a): where X (s, a) is the support of C(s, a).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We show that some off-the-shelf optimization methods can be used to obtain a solution ĥ close to a scaled and circularly shifted version of the ground truth.
Sentence 2: ĥ satisfies C f R ĥ ≈ ±e j for some j ∈ [n], i.e., R ĥ is approximately a signed and shifted version of the inverse of f . Given solution ĥ to (P1), one can recover f and x i (i = 1, 2, . . . , N ) as follows: Here, we use x −1 to denote the entrywise inverse of x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, since D is not a simple signal, but a more involved data structure, it is not clear how to run existing algorithms on D. In the supplementary material we show how to apply such algorithms on our coresets.
Sentence 2: we can run naive dynamic programming [4] on the coreset and get a (1 + ε) approximate solution in an efficient manner, as we summarize as follows.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we present theoretical analysis of SON -a convex optimization procedure for clustering using a sum-of-norms (SON) regularization recently proposed in [8,10,11,17].
Sentence 2: we show if the samples are drawn from two cubes, each being one cluster, then SON can provably identify the cluster membership provided that the distance between the two cubes is larger than a threshold which (linearly) depends on the size of the cube and the ratio of numbers of samples in each cluster.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that while they are based on the operations of the underlying Poisson processes, due to theorems 3, 5, and 7, we operate directly on the DPs, without the need of explicitly instantiating the associated Poisson processes or Gamma processes.
Sentence 2: let µ t be the base measure of D t , then if the acceptance probability q is a constant, then α t = qα t−1 + α ν .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consequently there exists a solution to the equation (6).
Sentence 2: we aim to find the solution closest to a given initial point x 0 2 X : Using the pseudoinverse and Lemma 22 item vi, the solution to ( 7) is given by where A † and A ⇤ denote the pseudoinverse and the adjoint of A, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The max-norm is defined as: Similarly to the rank, but unlike the trace-norm, generalization and learning guarantees based on the max-norm hold also under an arbitrary, non-uniform, sampling distribution.
Sentence 2: defining mc(X) = X 2 max (no normalization is necessary here), Õ(mc(X)(n + m)) samples are enough for generalization w.r.t.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The key contribution of this paper is to show that, for a certain class of PTL problems, the optimal common representation shared across infinitely many parameterized tasks can be obtainable.
Sentence 2: we develop an alternating minimization algorithm à la [1,2] for finding the entire continuum of solutions and the common feature subspace (or the common set of variables) among infinitely many parameterized tasks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Input: The base generative model p θ (x) The (set of) constraints f φ (x) 1: Initialize generative parameter θ and constraint parameter φ 2: repeat Connections to adversarial learning For implicit generative models, the two objectives w.r.t φ and θ (Eq.8 and Eq.12) are conceptually similar to the adversarial learning in GANs [15] and the variants such as energy-based GANs [26,55,54,50].
Sentence 2: the constraint f φ (x) can be seen as being optimized to assign lower energy (with the energy-based distribution q(x)) to real examples from p d (x), and higher energy to fake samples from q(x) which is the regularized model of the generator p θ (x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This produces a model very similar to previous machine learning work on "model-free" approximate POMDP solvers [5,6], which attempt to form good solutions without explicit knowledge of the environment's structure or dynamics.
Sentence 2: we model working memory as a discrete memory system (a collection of recent observations) rather than a continuous "belief state" (an inferred probability distribution over hidden states).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We randomly select m = 4sµ s = 160 data to form D m , out of with |Σ| = 2mr 2 = 5120 entries of the 160 × 160 sub-matrix are fed into the algorithm.
Sentence 2: roughly 0.5% entries out of the whole pairwise label matrix Z ∈ {0, 1} 1000×1000 are observed.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Hence we make the simplifying assumption that Pr(x, y) and Pr ′ (x, y) only differ via Pr(x, y) = Pr(y|x) Pr(x) and Pr(y|x) Pr ′ (x).
Sentence 2: the conditional probabilities of y|x remain unchanged (this particular case of sample selection bias has been termed covariate shift [12]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Namely we consider four different scenarios where the outliers are generated from models with different graphical structures.
Sentence 2: each sample is generated from the following mixture distribution: We also consider the scenario where the outliers are not symmetric about the mean and simulate data from the following model For each simulation run, Θ is a randomly generated precision matrix corresponding to a network with 9 hub nodes simulated as follows.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We apply this RL based interactive retrieval framework to the task of image retrieval, which we call dialog-based interactive image retrieval to emphasize its capability in aggregating history information compared to existing single turn approaches [13,14,15,16].
Sentence 2: a novel end-to-end dialog manager architecture is proposed, which takes natural language responses as user input, and delivers retrieved images as output.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since h ∈ H can either win a point (x, y, p) or lose it, the model resembles a binary classification task, where the "label" of (x, y, p) is the identity of the winner.
Sentence 2: a triplet (x, y, p) would be labeled 1 if the agent produced a better prediction than the opponent, and zero otherwise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each strategy variable D j has domain A ∪ j , and given its parent θ j , its CPD chooses an action from A ∪ j according to the mixed strategy σ θi→ai j .
Sentence 2: if j = i then Pr(D j = a j |θ j ) is equal to σ j (a j |θ j ) for all a j ∈ A j,θj and 0 for all a j ∈ A ∪ j \ A j,θj ; and if j = i we have Pr(D j = a i |θ j ) = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This minimax lower bound shows that we cannot hope for a better bound than (3) (up to log factors) if we only know the value of L * T . Second-order bounds Another type of improved regret bound was derived by Hazan and Kale [2011b] and involves a second-order quantity called the quadratic variation: where µ T = 1 T T t=1 t is the mean of all loss vectors.
Sentence 2: (Q T /T is the sum of the empirical variances of all the K arms).
Relationship: entailment
Domain: NeurIPS
Sentence 1: With q ij := q(y i = j) we define µ ij (q) = q ij for all i > m train and µ ij (q) = 1 if y i = 1 and 0 otherwise for all i ≤ m train .
Sentence 2: we are taking the expectation in µ over all unobserved labels Y test with respect to the distribution q(Y test ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: 3: For i = 2, 3, . . . , n perform: rithm, Λ i (s) represents the ordered intersection of the redundant-sets A j Cj up to the controller C i (i.e., 1 ≤ j ≤ i) constrained by the order of priority.
Sentence 2: each set Λ i (s) contains a set of actions in state s that are all i -ascending with respect to the superior controllers C 1 , C 2 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The KGV has an interesting relationship to the mutual information among the original variables, DB CB DB 1 .
Sentence 2: as shown in [6], in the case of two discrete variables, the KGV is equal to the mutual information up to second order, when expanding around the manifold of distributions that factorize in the trivial graphical model (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Compared with classical factorization Z V , such tri-factorization supports the asymmetric transitivity property of directed links.
Sentence 2: the existence of two directed links (i, j) (z i W z j ), (j, k) (z j W z k ) increase the likelihood of (i, k) (z i W z k ) via representation propagation z i → z j → z k , but not the case for (k, i) due to asymmetric W . Then we have a lemma as follows: Lemma 3.2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, the better approximation comes at a cost of increased computational complexity for both structure [4] and parameter learning [1] as compared to generative models.
Sentence 2: unlike Bayesian networks or junction trees [3], (a) the likelihood of a CRF structure does not decompose into a combination of small subcomponent scores, making many existing approaches to structure learning inapplicable, and, (b) instead of computing optimal parameters in closed form, with CRFs one has to resort to gradient-based methods.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To define the inverse of each of these operators, a reference must be made to the dimensionality of the original tensor.
Sentence 2: given X ∈ R I and A ∈ R IJ , where I, J ∈ N M , M ∈ N, we have imjm , where A (m) ∈ R Im×Jm for all m. The factorization exponentially reduces the number of parameters needed to express A from (1) , where ⊗ is the Kronecker matrix product [1].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Define a function, where We consider the following discriminative model; we model the log-odds ratio of the posterior class probability to be the sum of a bilinear function with respect to the EEG signal amplitude and linear with respect to the second order statistics of the EEG signal: The first term of the equation ( 1) can be interpreted as a spatio-temporal projection of the signal, under the bilinear model, and captures the first order statistics of the signal.
Sentence 2: the columns u r of U represent R linear projections in space (rows of X).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus, the beta part of the corresponding approximate factors can be removed and the multivariate Gaussian distributions simplify to univariate Gaussians.
Sentence 2: the approximate factors ψik with i = 1, . . . , n and k = y i are: where sik , pik , μik , νik , μyi ik and νyi ik are free parameters to be estimated by EP.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Hence, we here explore whether our model could still predict our data if attention might only modulate neuronal gain, but do so non-uniformly across neurons and tasks.
Sentence 2: we investigate whether modulating the gain of only the neurons that are loudest, best-tuned, or most informative about the stimulus, or of all neurons equally but in a task-dependent manner, may account for the data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The ith marginal of the final layer of the variational DGP posterior (12) depends only on the ith marginals of all the other layers.
Sentence 2: the consequence of this property is that taking a sample from q(f L i ) is straightforward, and furthermore we can perform the sampling using only univariate unit Gaussians using the 're-parameterization trick' (Rezende et al., 2014;Kingma et al., 2015).
Relationship: entailment
Domain: NeurIPS
Sentence 1: But any update to θ that tries to fit this value (i.e., makes )) forces θ1 > 0, which is inconsistent with the assumption, θ1 < 0, needed to generate the high bootstrapped value.
Sentence 2: any update that moves (s 2 , a 2 ) higher undercuts the justification for it to be higher.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This leads to a biased estimator.
Sentence 2: since what we care about is the generalization performance of the algorithm, a better method is to pick θ to maximize the log-likelihood of data that wasn't used to calculate the "features" b, because when we see a test example, we will not have had the luxury of incorporating information from the test example into the b's (cf.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, one dimensional algorithms cannot be easily extended to the multivariate case.
Sentence 2: the ideas of root bracketing and function transversal are not well defined [2]; given a particular bracket of a continuous surface, there will be an infinite number of solutions to the equation f ( x) − t = 0, since the solution in multiple dimensions is a set of surfaces, rather than a set of points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work, we propose a novel hybrid approach for tackling problems such as (3), that combines the ideas of the CHW and PST frameworks.
Sentence 2: we suggest using a SGD-like low-regret update for the variable w, while updating the variables ξ and b via a PST-like step; the dual update of our method is similar to that of CHW.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our model reproduced this behavior and provided a straightforward explanation: latent-to-observed weights (w ij ) in the MAP model were so strong that whenever a latent was switched on it could almost only produce triplets, therefore doublets were created by spontaneous independent activation of observeds which thus produced embedded and mixture doublets with equal chance.
Sentence 2: doublets were seen as mere noise under the MAP model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Contributions: (i) we propose a new algorithm to compute C(U ) exactly in O(q 2 p) steps, the first non-iterative algorithm to compute C(U ); (ii) we compare its performance against state-of-the-art iterative algorithms, and observe a much faster convergence.
Sentence 2: our algorithm scales much faster than O(q 2 p) in practice; (iii) we implement our algorithm on a GPU, and show that it computes the cost of all (more than 2 billion) trees with ≤ 11 nodes, in ≤ 2.5 hours.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Had we ignored the minimality requirement in definition 2 and chosen to factorize Y < X < Z < R y using the chain rule, we would have obtained: |Z does not hold in the graph.
Sentence 2: existence of one admissible sequence based on an order O of variables does not guarantee that every factorization based on O is admissible; it is for this reason that we need to impose the condition of minimality in definition 2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assume that P workers write updates at regular time intervals (referred to as "clocks").
Sentence 2: let u  where S p,c ✓ W p,c = ([1, P ] \ {p}) ⇥ [c s, c + s 1] is some subset of the updates u written in the width-2s "window" W p,c , which ranges from clock c s to c + s 1 and does not include updates from worker p. the noisy state xp,c consists of three parts: 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A Bayesian action-graph game (BAGG) is a tuple (N, Θ, P, {A i,θi } i∈N,θi∈Θi , G, {u α } α∈A ) where N is the set of agents; Θ = i Θ i is the set of type profiles; P is the type distribution, represented as a Bayesian network; A i,θi ⊆ A is the type-action set of i given θ i ; G = (A, E) is the action graph; and for each α ∈ A, the utility function is u α : C (α) → R. Intuitively, this representation captures two types of structure in utility functions: firstly, shared actions capture the game's anonymity structure: if two action choices from different type-action sets share an action node α, it means that these two actions are interchangeable as far as the other players' utilities are concerned.
Sentence 2: their utilities may depend on the number of players that chose the action node α, but not the identities of those players.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, with active oracle accesses, SVRG is also possible and the upper bound becomes: Our main contributions in this paper are: (1) presenting a precise formal oracle framework for studying parallel stochastic optimization; (2) establishing tight oracle lower bounds in this framework that can then be easily applied to particular instances of parallel optimization; and (3) using the framework to study specific settings, obtaining optimality guarantees, understanding where additional assumptions would be needed to break barriers, and, perhaps most importantly, identifying gaps in our understanding that highlight possibilities for algorithmic improvement.
Sentence 2: • For non-smooth objectives and a stochastic prox oracle, smoothing and acceleration can improve performance in the layer graph setting.
Relationship: entailment
Domain: NeurIPS
Sentence 1: n ∈ N the number of rows of Z, i.e.
Sentence 2: the number of customers sampling dishes from the " Indian Buffet", let m k be the number of customers who have sampled dish k, let K + be the total number of dishes sampled, and denote by K h the number of dishes with a particular selection history h ∈ {0; 1} n . K h > 1 only if there are two or more dishes that have been selected by exactly the same set of customers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Small-error linear filters Linear filter ϕ o with a small 1 norm in the spectral domain and small recovery error exists, essentially, whenever there exists a linear filter with small recovery error [8,4].
Sentence 2: indeed, let us say that x ∈ C(Z n 0 ) is simple [4] with parameters m ∈ Z + and ρ ≥ 1 if there exists x is (m, ρ)-simple if there exists a hypothetical filter φ o of the length at most m + 1 which recovers x τ with squared risk uniformly bounded by As proved in [15, Appendix C], we have and, for a moderate absolute constant c, with probability 1−α.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 1 To complete the reduced model, we need the conditional distributions P(G N j f N( j 1) ) and P( f N j f N( j 1) G N j ).
Sentence 2: how many "ones" do we expect in an interval, given the event status at the beginning of the interval, and what is the probability that an event is occurring at the beginning of the next interval, given also the number of ones in the current interval?
Relationship: entailment
Domain: NeurIPS
Sentence 1: The approximate inference algorithm aims to compute good approximations to the real posteriors by iteratively optimizing the above described variational bound.
Sentence 2: given the approximations q t (y i ) ∼ N (µ t yi , Σ t yi ) (similar forms for z i and w i ) and q t (α j i ) ∼ Γ(a t ij , b t ij ) from the t th iteration the update rules are as follows: Update for q t+1 (y i ): Alternating between the above described updates can be considered as message passing between the low-dimensional regression outputs and higher dimensional output labels, which in turn are constrained to be sparse.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We additionally study the total cost sufficient for learning, for an abstract notion of the cost of requesting the labels of a given number of examples at once.
Sentence 2: we find that for sublinear cost functions, it is often desirable to request labels in large batches (i.e., buying in bulk); although this may increase the total number of labels requested, it reduces the total cost required for learning.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our main result, Theorem 8, states that as the minipool size goes to infinity, E[∇ (z (t) , θ)] tends in the direction of the gradient of the population zero-one loss Z(θ), where the expectation is with respect to the randomness of the minipool X M and label y.
Sentence 2: let z (t) be chosen via uncertainty sampling with the parameters θ: ).
Relationship: entailment