sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: When f (A, w) ⌧ f ⇤ , CombCascade can distinguish A from A ⇤ without learning the expected weights of all items in A.
Sentence 2: combCascade acts implicitly on the prefixes of suboptimal solutions, and we choose them in our analysis such that the probability of observing all items in the prefixes is "close" to f ⇤ , and the gaps are "close" to those of the original solutions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While it is still common practice in both sparse coding and manifold learning to collect data in an orderless fashion, other work has used time-series data to learn spatiotemporal representations [57,41,30] or to disentangle form and motion [6,9,13].
Sentence 2: the combination of topography and temporal coherence in [30] provides a strong motivation for this work.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For subset selection under additive noise, POSS using By comparing Theorem 1 with 3, we find that the approximation bounds of POSS and the greedy algorithm under multiplicative noise are nearly the same.
Sentence 2: for the submodular case (where γ X,k = 1 for any X and k), they are exactly the same.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our study suggests that the filter properties of STDP originating from the dynamics of pre-and postsynaptic activity contributions are in fact exploited for learning in the brain.
Sentence 2: shifts in baseline rates, as well as the frequency and the respective phases of pre-and postsynaptic rate modulations induced by theta oscillations could be tuned to match the values that make STDP most susceptible for synaptic modifications.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Previously information theoretical approaches to neural coding have been used to quantify the amount of information that a neuron or a neural network is able to encode or transmit [6][7][8].
Sentence 2: algorithms based on the maximization of the mutual information between the output and the input of a network, also called infomax principle [9], have been used to detect the principal (or independent) components of the input signal, or to reduce the redundancy [10][11][12].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Hand and Voroninski [8] further establish that a fully connected generative network with weights following Gaussian distribution can be inverted given only compressive linear observations of its last layer.
Sentence 2: they show that under mild technical conditions Problem (3) has a favorable global geometry, in the sense that there are no stationary points outside of neighborhoods around the desired solution and its negative multiple with high probability.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2 can be rewritten as L π = p 0 r 0 + c/r 1 , where r i is the mean firing rate when the inputs are generated from f i , and p 0 is the fraction of time when f 0 is applicable (as opposed to f 1 ).
Sentence 2: if the rate of change is slow compared to neuronal firing rates, then optimal processing amounts to minimizing the "spontaneous" firing rate during f 0 and maximizing the "stimulus-evoked" firing rate during f 1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: For that purpose, define a matrix Φ(s, s) = 1 s=θ(s) Algorithm 2: Solve (3.1) in Line 15 of Algorithm 1 Input: z s , q s -sorted such that z s is non-decreasing, indexed as 1 . . . n Output: ξ s,a -optimal solution of (3.1) // Determine the threshold 5 do where s ∈ S, s ∈ S, and 1 represents the indicator function.
Sentence 2: each column corresponds to a single aggregate state with each row entry being either 1 or 0 depending on whether the original state belongs into the aggregate state.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we start our experiment with triple pole-balancing using exactly the same configuration, and then we let KBSF refine its model M by incorporating more sample transitions through update rules (3) and ( 4).
Sentence 2: we used Algorithm 2 with a 0.3-greedy policy, t m = t v = 10 6 , and n = 10 7 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The corresponding sub-counts n vjk can be viewed as the result of the contribution of underlying biological process k to the expression of gene v in sample j of domain d. The probability parameter p (d) j , which depends on the sample index, accounts for the potential effect of varying sequencing depth of sample j in domain d. More precisely, the expected expression of gene v in sample j and domain d is , and hence the term kj ) can be viewed as the true abundance of gene v in domain d, after adjusting for the sequencing depth variation across samples.
Sentence 2: it comprises of contributions from both domain-dependent and globally shared latent factors, where the amount of contribution of each latent factor can automatically be learned for the sample relevance across domains.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We define U (I) to be the maximum payoff reachable from a history in I, and L(I) to be the minimum.
Sentence 2: u (I) = max z∈Z,h∈I:h z u P (I) (z) and L(I) = min z∈Z,h∈I:h z u P (I) (z).
Relationship: entailment
Domain: NeurIPS
Sentence 1: A special case of MDP under which PSRL has been recently extensively studied is MDP with state resetting, either explicitly or implicitly.
Sentence 2: in [Osband et al., 2013;Osband and Van Roy, 2014] the considered MDPs are assumed to have fixed-length episodes, and at the end of each episode the MDP's state is reset according to a fixed state distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is sensible since, by the design of a MSR, trader i's belief-dependent action influences the action of each of (rational) traders i + 1, i + 2, . . . so that the action of each of these successors, in turn, has a role to play in determining the market impact of trader i's belief.
Sentence 2: if τ j = τ > 0 ∀j ≥ 1, then the aggregation weights satisfy the inequalities α n j+1 / α LMSR assigns progessively higher weights to traders arriving later in the market's lifetime when they all exhibit identical constant risk aversion.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since the rank function is neither differentiable nor convex, its relaxation such as Schatten-p norm has been used in respective optimization formulations.
Sentence 2: we consider the smoothed nuclear norm (i.e., Schatten-1 norm) minimization [18,20] that corresponds to ] d×r is a given matrix with missing entries, Ω indicates the positions of known entries and λ is a weight parameter and ε > 0 is a smoothing parameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The driving force behind such results are the isoperimetric inequalities which can be extended from uniform to general logconcave distributions.
Sentence 2: computing the volume of a convex body can be seen as a special case of integration of a logconcave function, and there has been a number of major results on mixing time for sampling from logconcave distributions [17,18].
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , A k } denote the partition induced by T . Let j(A) denote the depth of A and note that λ(A) = 2 −j(A) where λ is the Lebesgue measure on R d . Define T to be the collection of all DDTs and A to be the collection of all cells corresponding to nodes Let M be a dyadic integer, that is, M = 2 L for some nonnegative integer L. Define T M to be the collection of all DDTs such that no terminal cell has a sidelength smaller than 2 −L .
Sentence 2: no coordinate is split more than L times when traversing a path from the root to a leaf.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition, we assume the existence of a set of k A equality and k B inequality constraints on the random variables, that is, A(x) = a, where A : D → R k A , a ∈ R k A and B(x) ≤ b, where Both equality and inequality constraints restrict the possible combinations of values the random variables X can assume.
Sentence 2: we set f (x) = 0 whenever any of the constraints are violated and constrain the domain of integration, denoted D, for the normalization constant correspondingly.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, when X's are frequent but Y is presented, individuals are predisposed toward producing the X response, and this predisposition must be overcome by the perceptual evidence from the Y. Jones and Braver (2001) also performed an fMRI study of this task and found that anterior cingulate cortex (ACC) becomes activated in situations involving response conflict.
Sentence 2: when one stimulus occurs infrequently relative to the other, event-related fMRI response in the ACC is greater for the low frequency stimulus.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our problem, this translates to permitting the generating process to have an unrestricted number of latent components, but requiring that only a small number of them contribute to the composition of the histogram represented by any data instance.
Sentence 2: the latent components must be learned such that the mixture weights with which they are combined to generate any data have low entropy -a set with low entropy implies that only a few mixture weight terms are significant.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our notion of margin ρ in (ρ, ∆, ε)-separated instances is analogous to the margin separation notion used by the above results on spectral clustering [15,6].
Sentence 2: we require a margin of ρ = Ω(∆/ε 2 ) where ∆ is our scale parameter, with no extra √ k factor.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, this can be computed by finding the parameters of an OU process that matches q 0 in the moment matching Kullback-Leibler divergence.
Sentence 2: if q * ({x t }) minimises D[q 0 ({x t })||q * ({x t })], then the parameters of q * are given by where m bw t and V bw t are computed by the backward Kalman-Bucy filtering equations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus, we need to minimize (f 1 , g) + Ω(f 1 , g) with respect to f 1 and g. Since both f 1 and g are gradient boosted trees, we naturally adopt a stage-wise approximation for the objective.
Sentence 2: we define an impurity function which on the one hand approximates the negative gradient of (f 1 , g) with the squared loss, and on the other hand penalizes the initial acquisition of features by their cost c α .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Temporal frequencies were chosen so that the set of cells at each spatial frequency had the same pattern of velocity tunings.
Sentence 2: the sets tuned to low (0.0625 cycles/pixel), medium (0.125 cycles/pixel), and high (0.25 cycles/pixel) spatial frequencies had velocity tunings of 0.25, 0.5, 1.0, 2.0, and 4.0 pixels per time frame.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let S u,s denote the set of vertices which are at distance s from vertex u in the graph defined by edge set E 1 .
Sentence 2: i ∈ S u,s if the shortest path in G 1 = ([n], E 1 ) from u to i has a length of s. Let T u denote a breadth-first tree in G 1 rooted at vertex u.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since n 1 n 2 50 for all patient sets, our collection of twosample problems is genuinely high-dimensional.
Sentence 2: we have 14 × ( 6 2 + 4 2 + 6 2 ) = 504 problems under H 0 and 14 × (6 • 4 + 6 • 4 + 6 • 6) = 1176 problems under H 1 -assuming that every gene set was differentially expressed between two sets of patients with different cancers, and that no gene set was differentially expressed between two sets of patients with the same cancer type.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given the output q of Policy 1, we obtain an approximate Q-value for each (continuous) state-action pair (x, a) 2 Z via the nearest-neighbor average operation, i.e., Q T h (x, a) = ( NN q) (x, a); here the superscript T emphasizes that the algorithm is run for T time steps with a sample size of T . As a main result of this paper, we obtain finite-sample analysis of NNQL policy.
Sentence 2: we find that the NNQL policy converges to an "-accurate estimate of the optimal Q ⇤ with time T that has polynomial dependence on the model parameters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we break down problem (1) into element-wise product, it becomes: where A ij:ab corresponds to the edge similarity between edge Here H ∈ {0, 1} 2n×n 2 is a selection matrix over the elements of x sufficing assignment constraints according to (1).
Sentence 2: we relax x into continuous domain and let f prod (x ia , x jb ) = x ia x jb : We generalize problem (3) by replacing f prod with f δ : where f δ is a 2D quasi-delta function in the continuous domain (f δ (x, y) = 1 if x = 1 and y = 1, and f δ = 0 otherwise).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The resulting classifier defines a nonlinear separating boundary in the original feature space.
Sentence 2: equations ( 1) and (2) imply that the classification function f K (x) is directly proportional to the signed distance from the input point to the separating boundary computed in the higher dimensional space defined by the mapping Φ K . the function output depends only on the projection of vector Φ K (x) onto w and completely ignores the component of Φ K (x) that is perpendicular to w. This suggests that in order to create a displacement of Φ K (x) that corresponds to the differences between the two classes, one should change the vector's projection onto w while keeping its perpendicular component the same.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We used posterior predictive checking to assess the extent to which our model is a "good fit" for the NHC email network [16,17].
Sentence 2: we defined four network statistics (i.e., four discrepancy functions) that summarize meaningful aspects of the NHC network: generalized graph transitivity, the dyad intensity distribution, the vertex degree distribution, and the geodesic distance distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In fact, we could make the constant arbitrary small for large enough T . In our experiments we focused on pipelined optimization.
Sentence 2: we used two different training sets that were based on e-mails: the TREC dataset [5], consisting of 75,419 e-mail messages, and a proprietary (significantly harder) dataset of which we took 100,000 e-mails.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One useful result from their analysis is a derivation of the optimal schedule along a given path.
Sentence 2: the value of F(γ) using the optimal schedule is given by (γ) 2 /2, where is the Riemannian path length defined by Intuitively, the optimal schedule allocates more distributions to regions where p β changes quickly.
Relationship: entailment
Domain: NeurIPS
Sentence 1: More formally, a tensor X ∈ R I1×•••×I M is a multidimensional array with elements that can each be indexed by a vector of positive integers.
Sentence 2: every element X i1•••i M ∈ R is uniquely addressed by a vector (i 1 , • • • , i M ) such that 1 ≤ i m ≤ I m for all m. Each of the M dimensions of X is called a mode and represents a particular component of the data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We extend the above strategy by explicitly representing the norm of VQ residuals, learning a PQ codebook only on the unit-normalized rotated VQ residuals, while separately scalar quantizing the residual norm scales.
Sentence 2: multiscale quantization employs the following steps: (1) vector quantization of the dataset; (2) learned rotation of the vector quantization residuals; (3) reparameterization of the rotated residuals into direction and scale components; (4) product quantization of the direction component; (5) scalar quantization of the scale component.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The main advantage of the patch-based approach is that object-detection can be reduced to a binary classification problem.
Sentence 2: we compute P (O c i = 1|v c i ) for each class c and patch i (ranging over location and scale), where O c i = 1 if patch i contains (part of) an instance of class c, and O c i = 0 otherwise; v c i is the feature vector (to be described below) for patch i computed for class c. To detect an object, we slide our detector across the image pyramid and classify all the patches at each location and scale (20% increments of size and every other pixel in location).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Concretely, let f (i) j represent the j th feature value (rows) for the i th example (columns), where f . Our method simply involves first normalizing the feature distribution matrix by rows, then by columns and finally summing up the absolute value of all entries.
Sentence 2: we first normalize each feature to be equally active by dividing each feature by its 2norm across all examples: fj = f j / f j 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The model parameters, θ, are specified as By setting W 3 and W 4 to be zero matrices, the TSBN can be viewed as a Hidden Markov Model [1] with an exponentially large state space, that has a compact parameterization of the transition and the emission probabilities.
Sentence 2: each hidden state in the HMM is represented as a one-hot length-J vector, while in the TSBN, the hidden states can be any length-J binary vector.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Figure 3 shows that when using fixed code length K, as the Super-Bit depth N gets larger (1 < N  min(d, K)), the MSE of SBLSH gets smaller, and the gap between SBLSH and SRP-LSH gets larger.
Sentence 2: when N = K, over 30% MSE reduction can be observed on all the datasets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: By studying the ODE, we establish the global convergence of Async-MSGD, and the rate of convergence will be established later.
Sentence 2: we consider a continuous-time interpolation V ⌘,⌧ (t) of the solution trajectory of the algorithm: For t 0, set V ⌘,⌧ (t) = v ⌘,⌧ k on the time interval [k⌘, k⌘ + ⌘).
Relationship: entailment
Domain: NeurIPS
Sentence 1: For each i ∈ I, let us collect the cliques that i participates in: define sets We show that u can be feasible if and only if u Ji + u Ki ≥ 1 for all i ∈ I, where u S denotes a product.
Sentence 2: u can be feasible if and only if, for each i, (⇒) If u j ≤ x Pj x Qj for all j ∈ V, then having u Ji + u Ki ≥ 1 is necessary: if both u Ji = 0 and u Ki = 0 for any i it would mean there exists j ∈ J i and k ∈ K i for which x Pj = 1 and x Q k = 0, contradicting any unique assignment to x i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Modeling catching as a continuous partially observable Markov decision process and employing stochastic optimal control theory, we discover that the four main heuristics described in the literature are optimal solutions if the catcher has sufficient time to continuously visually track the ball.
Sentence 2: by varying model parameters such as noise, time to ground contact, and perceptual latency, we show that different strategies arise under different circumstances.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Further, our converse results in Section 2.3 provides a fundamental limit on the achievable error rates under differential privacy.
Sentence 2: with data collected from an ε-locally differentially privatization mechanism, one cannot achieve an asymptotic type II error smaller than 2(e ε + 1) D kl (P 0 ||P 1 ) , whenever ε ≤ ε * , where ε * is dictated by Theorem 2.3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will assume that S i ∩ S j = ∅ for all i, j and that ∪ i S i = {1, . . . , N }.
Sentence 2: block coordinate minimization algorithms work as follows: at each iteration, first set δ t+1 = δ t . Next choose a block S i and set: where we use F i (δ Si ; δ t ) to denote the function F restricted to the variables δ Si and where all other variables are set to their value in δ t . at each iteration we fully optimize only over the variables δ Si while fixing all other variables.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition, we expect that the data points in each cluster be associated with the representatives in that cluster only.
Sentence 2: definition 1 Given dissimilarities {d ij } i,j=1,...,N between N data points, we say that the data partitions into n clusters {C i } n i=1 according to the dissimilarities, if for any data point j in any C j , the largest dissimilarity to other data points in C j is strictly smaller than the smallest dissimilarity to the data points in any C i different from C j , i.e., max i ∈Cj the data partitions into clusters {C i } n i=1 , when the interclass dissimilarity is smaller than the intraclass dissimilarity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From the second order Taylor expansion with Lagrangian remainder and the assumption that is twice differentiable, it follows that there exists a constant C( θ * , D) depending only on the local minimizer θ and the dataset D such that the following inequality holds for every sample in the dataset and every ε ∈ [0, ε 0 ), Summing the above inequality over all samples in the dataset and recalling that Ln ( θ * +∆) ≥ Ln ( θ * ) holds for all ε ∈ [0, ε 0 ), we obtain Finally, we complete the proof by induction.
Sentence 2: for the base hypothesis where p = 0, we can take the limit on the both sides of the above inequality as ε → 0, using the property that δ a can be either positive or negative and thus establish the base case where p = 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the overlap case (γ < 0), supervised learners are always limited by the error incurred due to averaging across decision sets (n −1/d ).
Sentence 2: for the collection of distributions with γ < −γ 0 , a faster rate of error convergence is attained by SSL compared to SL, provided m ≫ n 2d .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, GDMM-subFMO achieves orders-of-magnitude speedup over other methods.
Sentence 2: when running on ChineseOCR and EUR-Lex, each iteration of SSG, GDMM, BCFW and Soft-BCFW take over 10 3 seconds, while it only takes a few seconds in GDMM-subFMO.
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, we believe that our model's unusual compatibility with larger, less manicured data sets will prove to be a major strength moving forward.
Sentence 2: the skip-gram language model we constructed included only a modestly sized vocabulary, and was exposed only to the text of a single online encyclopedia; we believe that the gains available to models with larger vocabularies and trained on vastly larger text corpora will be significant, and easily outstrip methods which rely on manually constructed semantic hierarchies (e.g.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A later datum stays at node with probability (N +1)/(N +N ≺• +α(| |)+1), where N is the number of previous data that stopped at , and N ≺• is the number of previous data that came down this path of the tree but did not stop at , i.e., a sum over all descendants: N ≺• = ≺ N . If a datum descends to but does not stop then it chooses which child to descend to according to a Chinese restaurant process where the previous customers are only those data who have also descended to this point.
Sentence 2: if it has reached node but will not stay there, it descends to existing child i with probability (N i +N i≺• )/(N ≺• +γ) and instantiates a new child with probability γ/(N ≺• +γ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Of course, any problem can be embedded into the complete graph by setting to zero a subset of the {θ st } parameters.
Sentence 2: (for a graph G = (V, E), we simply set θ st = 0 for all pairs (s, t) / ∈ E.) We first focus on the marginal polytope MARG(K n ) ≡ MARG(K n ; φ) of valid dual variables {µ s , µ st }, as defined in Eqn.
Relationship: entailment