sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: 16 for power-law decay in the ridge leverage scores when, where C is an absolute constant.
Sentence 2: one advantage of using deterministic rather than randomized rank-k subspace leverage score algorithms is that under the condition of power-law decay in the sorted rank-k subspace leverage scores, the deterministic algorithm chooses fewer columns than random sampling with the same error for the column subset selection bound when max((2k/ ) , where a is the decay power and C is an absolute constant (Papailiopoulos et al., 2014) (this is the counterpart to our Theorem 5).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since the array contains empty entries, the ratings were shifted by -3.5 to have the possible ratings {-2.5, -1.5, -0.5, 0.5,1.5, 2.5} which allows to assign the value of zero to the empty entries of the array (movies which were not rated).
Sentence 2: all the thresholds aj, bj can be recovered from the support vectors -unlike the fixed-margin scheme which required another LP pass.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This research is supported in part by Grants (CUHK 14206715 & 14222816) from the Hong Kong RGC, the Major Research Plan of the National Natural Science Foundation of China (Nos.
Sentence 2: for the specific problem in (9) with the inner product in (10), we can derive a simpler form to solve Y k below.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Consider a curve γ(s) on bd(W) connecting w (1) and w (2) that is defined by the intersection of bd(W) and P and is parametrized by its curve length s so that γ(0) = w (1) and γ(l) = w (2) , where l is the length of the curve γ between w (1) and w (2) .
Sentence 2: for −Θ t , −Θ t−1 ∈ V , w t+1 = w t . Hence, by Theorem 3.9, On the other hand, note that f i ∞ ≤ M . Then where the last inequality is due to Hoeffding's inequality.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Rather than counting the number of times each of the n words are separately manifested, we may more efficiently count the number of times words in particular subsets of D are manifested.
Sentence 2: in this context, for each draw X there is a latent class random variable C ∈ {1, . . . , L}, where the probability of class c is π c . Our goal is to design Φ such that the observed Y is most informative about the underlying X or C. When the interest is in recovering X, we design Φ with the goal of maximizing mutual information I(X; Y ), while when interested in inferring C we design Φ with the goal of maximizing I(C; Y ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that if we examine the posterior distributions of the weights we see that if each x n contains only one nonzero value then each factor in the likelihood is dependent on only one of the K weights and can be multiplied into the corresponding k th spike-and-slab.
Sentence 2: in our experimental setup, at each trial, n = 1, . . . , N , the experimenter stimulates R of K possible presynaptic neurons.
Relationship: neutral
Domain: NeurIPS
Sentence 1: If the dimensionality gets so large that maintaining a full covariance matrix becomes computationally infeasible, one has to resort to low-dimensional approximations [e.g., Loshchilov, 2015], which, however, bear the risk of a significant drop in optimization performance.
Sentence 2: in contrast to previous attempts in this direction, we present an approach that comes with a theoretical justification for why it does not deteriorate the algorithm's performance.
Relationship: neutral
Domain: NeurIPS
Sentence 1: It was also necessary to adjust the sampling of the nonlinearities (location of µ jk 's) because, as the fixed-norm filters rotated through input space, the variance of the projections can change drastically.
Sentence 2: measured neural nonlinearities are typically softer, but when rectified noise is taken into account, a hard-rectified model has been shown to be a good description of neural variability [32].
Relationship: neutral
Domain: NeurIPS
Sentence 1: By concatenation of the vectors n r (r = 1, . . . , 20) containing the bins of each neuron we obtain one data point x = [n 1 n 2 ... n 20 ] per trial.
Sentence 2: hence, the feature map is defined as: each trial is represented by a vector of its alignment score with respect to all other trials where alignments are computed separately for all 20 neurons.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our metric includes the existing fundamental metrics for dynamical systems, which are basically defined with principal angles between some appropriately-chosen subspaces, as its special cases.
Sentence 2: a variant of estimation algorithms, called dynamic mode decomposition (DMD), has been successfully applied in many real-world problems, such as image processing [11], neuroscience [3], and system control [16].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Independence given the label would say that given that x 1 is in some positive cluster C i in X 1 , x 2 is equally likely to be in any of the positive clusters C j in X 2 .
Sentence 2: since the stationary distribution of this walk can be viewed as having equal probability on every edge, this is equivalent to saying that for any partition of the graph into two pieces (S, V − S), the number of edges crossing the partition should be at least an fraction of the number of edges in the smaller half.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Consider the least squares estimator dè` ¡ 3f gha of , where iè` ¡ 'f is the pseudo-inverse of h` .
Sentence 2: after tokenization we lemmatized the tokens using a general-purpose finite-state morphological English analyzer and then removed stopwords (we also replaced all digits with a single special character).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that synthetic dataset II is more challenging than synthetic dataset I, because the smallest magnitude of Π * in synthetic dataset I is 0.2, while that in synthetic dataset II is much smaller (about 10 −3 ).
Sentence 2: under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-k, and attains a s/n statistical rate of convergence with s being the subspace sparsity level and n the sample size.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For any (potentially non-evaluated) design x ∈ X , the posterior mean µ k (x) and posterior variance σ 2 k (x) of the GP, conditioned on S k , are known in closed-form and are considered cheap to evaluate: where is the k × 1 vector whose i th entry is κ(x i , x) (respectively y i ), and λ is the noise variance.
Sentence 2: it requires the solution of nested maximizations and expectations for which there is no closed-form expression known.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Numerical evaluations of bias and variance of estimators discussed in the present section only rely on the computation of coefficients θ α,β defined by where α, β are non-negative integers and (Z, Z ) are bivariate normal (1) with r = ρ.
Sentence 2: the papers [25,32] discuss quantized RPs with a focus on image retrieval based on nearest neighbor search.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The inner product of ( 9) is the margin of g(x) after this class reweighting.
Sentence 2: in the multiclass setting, this is only part of the weighting mechanism, since the terms ρ k (x i ) of ( 9)-( 10) are coefficients of a soft-min operator over margin components F (x i ), y zi − y k . Assuming the soft-min closely approximates the min, (9) becomes where and y is the codeword of ( 11).
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the adaptive Spike Response Model (aSRM), the dynamics of the (normalized) membrane-potential V (t) are described as a sum of integrated input current I(t) and spike-triggered currents η(t): where {t i } denotes the set of past emitted spikes, and the kernel φ(t) is a fast exponential low-pass filter on membrane currents: with τ m determined by the membrane capacitance and conductance, and is typically on the order of several milliseconds [23,17] .
Sentence 2: this is shown in the light grey (scaled additive aSRM) and dark grey (scaled non-adaptive SRM) bars in figure 6b.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This compares to -86.22 from [24] and -84.62 for a Deep Boltzmann Machine from [23]; however, we caution that these numbers no longer reflect true performance on the test set due to bias from AIS and repeated overfitting [23].
Sentence 2: combining the adaptive nature of RMSprop with non-Euclidean optimization provides dramatically improved performance, seemingly converging faster and to a better optimum.
Relationship: neutral
Domain: NeurIPS
Sentence 1: b * ( w) = J( w)) and (2) higher values of σ are desirable, as they reduce φ by increasing the size of its denominator.
Sentence 2: model-free methods are particularly exciting in this domain because direct numerical simulation can take days (Shelley et al., 2005) -in contrast optimizationg on the experimental physical flapping wing can be done in real-time, at the cost of dealing with noise in the evaluation of the cost function; success here would be enabling for experimental fluid dynamics.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Interestingly, the identities of the nations commanding the highest prices (in order, U.S., France, Switzerland, Germany, Italy, Spain, Netherlands) overlaps significantly with the k = 3 case, suggesting a certain robustness in the relative economic status predicted by the model.
Sentence 2: we thus will also consider a variant of this model in which at each time step, a new seller is still attached to exactly one extant buyer, while each new buyer is connected to ν > 1 extant sellers.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The USPS data set consists of 11,000, 16 × 16 grayscale images of handwritten digits (1, 100 images of each digit 0 through 9) with no fixed division.
Sentence 2: we can apply the PCT to herding by identifying It is now easy to see that, in general, herding does not converge because under very mild conditions we can always find an x * t such that w T t v t < 0.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We investigated how model performance varied as a function of training data, and found that LN models were more susceptible to overfitting than CNNs, despite having fewer parameters (Figure 4A).
Sentence 2: historically, much of our understanding of the retina comes from fitting models to responses to artificial stimuli and then generalizing this understanding to cases where the stimulus distribution is more natural.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The learner has Θ = {θ A , θ B }, and we want to teach it the fact that the world is using θ * = θ A .
Sentence 2: this is sensible: in fact, one can show that the minimum teaching set for a D-dimensional Gaussian is the D + 1 points at the vertices of a D-dimensional tetrahedron.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Certain types of stochastic search methods, such as Markov Chain Monte Carlo (MCMC) or simulated annealing, can be proven to be consistent with respect to the identification of a structure maximizing posterior probability [4,5,6,7].
Sentence 2: for each pair of clique candidates c and c such that c ⊂ c , ¬x c ∨ ¬x c is a logical consequence of the constraints (4).
Relationship: neutral
Domain: NeurIPS
Sentence 1: That is, after observing infinite many samples, streaming EM converges to the true means and matches the corresponding offline results in [3,6]; to the best of our knowledge this is also first such consistency result for the streaming mixture problem.
Sentence 2: the second shows that if the candidate centers are sufficiently close to the true centers, then the failure probability of mis-classifying a point to a wrong center is (upto constant factors) the probability of mis-classification even in the optimal solution (with true centers).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We remark that Markov Random fields have become a defining methodology in stereo reconstruction [15], along with layered EM-style methods [2,16]; see the comparison in [14].
Sentence 2: those techniques differ in characteristics from laser-based system to an extent that renders them practically inapplicable for many applications (most notably: long-range acquisition, where time-of-flight techniques are an order of magnitude more accurate then triangulation techniques, and bright-light outdoor environments).
Relationship: neutral
Domain: NeurIPS
Sentence 1: for the examples shown on Figure 2 (e), (f) we get I(x, y) ≈ 1.03 and I(x, y) ≈ 1.10 respectively).
Sentence 2: in contrast to most applications of the infomax principle (Linsker (1988)) in stochastic channels (e.g.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Swimmer-5 Swimmer-3 Swimmer-3 Figure 3: Left: For a 3-link swimmer, with relatively simple dynamics, the compared methods yield similar results and possibly a slight advantage to the purely model-based SVG (1).
Sentence 2: we were able to learn gripping and walking behavior, although walking policies that achieved similar reward levels did not always exhibit equally good walking phenotypes.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that this notion of neighbors for DME in the context of distributed gradient descent translates to two data sets F = f 1 , f 2 , .
Sentence 2: there are many subsequent works e.g., see [21] and in particular [5] showed that stochastic quantization and Elias coding [15] can be used to obtain communication-optimal SGD for convex functions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Retrieval metrics B-1 B-2 B-3 B-4 CIDEr METEOR R@1 R@5 R@10 MedRank New York City (CNN+LSTM) [30]  similarity metrics (BLEU) and retrieval metrics (R@K, median Rank).
Sentence 2: when we perform the LSA-based text summarization, we select top-κ highest ranked summary sentences, among which the top-ranked one becomes the summary sentence for the associated image, and all the top-κ ones are used for training in our model.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Considering the case of k = n, i.e., maximal possible induced matchings that are contained in a perfect matching (that is a matching of cardinality n), we show that if a d-regular graph is an Our upper bound on α(n) establishes an inherent limitation on the multitasking capacity of any network.
Sentence 2: first, our model is graph-theoretic in nature, focusing exclusively on the adjacency structure of tasks and does not consider many parameters that emerge in biological and artificial parallel architectures.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This expectation over S, Z, Z of this Gibbs average is simplified using integration by parts over the Gaussian noise z iµjν , z iµjν and Nishimori identities (see e.g.
Sentence 2: on one hand the shifted profile is a small perturbation of E s which matches a fixed point, except where it is constant, so if we Taylor expand, the first order vanishes and the second order and higher orders can be estimated as |i w,L (S(E s ); ∆)−i w,L (E s ; ∆)| = O(1/w) uniformly in L. by explicit cancellation of telescopic sums i w,L (S(E s ); ∆)−i w,L (E s ; ∆) = i RS (E good ; ∆)−i RS (E ∞ µmax ; ∆).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Similarly, as the shapes of observed objects change over time (because the only measurement in laser is depth), morphology based object tracking algorithms and optical flow [11,12] which are commonly used in computer vision are unsuitable.
Sentence 2: in spite of its profound formulation, it is less pragmatic for online learning due to O(N 3 ) computational cost in both learning and inference, where N is the number of data points.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In our application such structural information is the microeconomic information that dictates what agent behaviors are successful in a given policy and how these behaviors evolve over time.
Sentence 2: what needs to be estimated is the causal effect of a policy change, which is defined as the difference between the objective value when the economy is treated, i.e., when all agents interact under the new rules, relative to when the same economy is in control, i.e., when all agents interact under the baseline rules.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , x n ), where each variable x i ∈ F p (F p is the set of all natural numbers between zero and p − 1, for a given prime p) and g : F n p → F p . The prover P seeks to prove the following claim: that is, the sum of g evaluated at 2 n points is y. P and V now engage in a sum-check protocol to verify this claim.
Sentence 2: the inputs, weights and all intermediate values computed in the network must lie in this range.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Coming back to the problem of MIPS, if Q and P are different, the event {h(Q(x)) = h(P (x))} will not have probability equal to 1 in general.
Sentence 2: definition: (Asymmetric Locality Sensitive Hashing (ALSH)) A family H, along with the two vector functions Q ∶ R D ↦ R D ′ (Query Transformation) and P ∶ R D ↦ R D ′ (Preprocessing Transformation), is called (S 0 , cS 0 , p 1 , p 2 )-sensitive if, for a given c-NN instance with query q and any x in the collection S, the hash function h chosen uniformly from H satisfies the following: When Q(x) = P (x) = x, we recover the vanilla LSH definition with h(.)
Relationship: neutral
Domain: NeurIPS
Sentence 1: [ER60] Let C be a tight community with |C| = !
Sentence 2: similarly as for the stochastic block model, edge probabilities for the independent cascade model may vary between different communities and are identical within a single community C, where all edges have weights q ic C . an edge e between two nodes in a community C is in E with probability p C := q ic C • q sb C , independently for every edge, where q ic C and q sb C are the edge probabilities in the independent cascade model and the stochastic block model respectively.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The first metric is value unfairness, which measures inconsistency in signed estimation error across the user types, computed as where E g [y] j is the average predicted score for the jth item from disadvantaged users, E ¬g [y] j is the average predicted score for advantaged users, and E g [r] j and E ¬g [r] j are the average ratings for the disadvantaged and advantaged users, respectively.
Sentence 2: we consider the task of collaborative filtering using matrix factorization [19].
Relationship: neutral
Domain: NeurIPS
Sentence 1: when the alignment of predicted and ground truth is lost and the model does not have the power to restore it, more advanced optimization methods like dynamic time warping (Müller, 2007) during the matching phase may help at the cost of the simplicity and seamless integration of the scheduled sampling, as described in Section 3.
Sentence 2: there may be a more interesting interplay between ideal skip intervals and switching points for options, which suggest that they should ideally be learned jointly.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In particular, for t large enough, Newton's method yields k ✓t+1 ✓ ⇤ k 2  ⇠ 2,NM k ✓t ✓ ⇤ k 2 2 , and it is insensitive to the condition number of the Hessian.
Sentence 2: if a uniform sub-sample is used, the sub-sampled Hessian will be a random matrix with expected value at the true Hessian, which can be considered as a sample estimator to the mean.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Gal and Ghahramani proposed an approach based on dropout [4] to approximately sample from a posterior over neural networks.
Sentence 2: let c t,a = |{τ ≤ t : A τ = a}| be the number of times that action a has been selected by time t. We apply a coupling argument that introduces dependencies between the noise terms W t and action counts, without changing the distributions of any observable variables.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Early work on MAB problems (Robbins, 1952;Lai and Robbins, 1985;Gittins et al., 2011) tended to be more focused on asymptotic guarantees, whereas more recent work (Auer et al., 2002;Auer, 2003) has been directed towards a more "finite time" approach in which we can bound regret for fixed time horizons T . One of the best-known and well-studied techniques is known as the Upper Confidence Bound (UCB) algorithm (Auer et al., 2002;Auer and Ortner, 2010).
Sentence 2: for a given threshold index j, KMUCB tracks the arm elimination process as follows: for any threshold values below j, KMUCB believes that we have determined the best arm, and plays that arm constantly.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The AP algorithm addresses this issue by that each sample is initially viewed as an examplar and then examplar-to-member and member-to-examplar messages competitively transmit among all samples until a group of good examplars and their corresponding clusters emerge.
Sentence 2: the global descriptor ε C and the popularity character χ C are not sensitive to the local data scale at each point, leading to the robustness of Zeta merging against the variation of data scales.
Relationship: neutral
Domain: NeurIPS
Sentence 1: S pecifically, we assume exponentially decaying noise correlation with As this model is invariant under spatial shifts, the first PRF can be calculated by finding the spatial frequency at which the S NR is maximal.
Sentence 2: if a hypothetical neuron receives input from the population, and wants to decode the presence of the stimulus a 1 , the weights of the optimal linear readout [ 1 6] could be derived from b 1 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: If the length of X a..b is linear in n, so that b − a = αn for some α ∈ (0, 1), then its intra-subsequence distance ∆ x (a, b) (defined by ( 7)) converges to 0 with n going to infinity.
Sentence 2: experiment 2: (Dependence on λ) Algorithm 1 requires λ ∈ (0, 1) as a lower-bound on λ min .
Relationship: neutral
Domain: NeurIPS
Sentence 1: As a common practice [28], assume that (t, y) is Lipschitz continuous in t for all |t| ≤ C g with a Lipschitz constant L . Theorem 4 (Estimation error bound).
Sentence 2: for 20News, we borrowed the pre-trained word embeddings from GloVe [42], and the model can be written as d-avg_pool(word_emb(d,300))-300-300-1,  ; it is one of the best architectures for CIFAR-10.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Therefore, finding the shortest path on the graph is equivalent to minimizing the total loss, which is the aim in loss-based decoding.
Sentence 2: finally, we note that although PD-Sparse [15] and DiSMEC [2] perform better on some model size regions of the datasets, their worse case space requirement during training is linear in the number of classes K, whereas our approach guarantees (adjustable) logarithmic space for training.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The obvious choice is the so-called defining representation, whose elements are the familiar permutation matrices ρ def (σ) = P (σ) [P (σ)] p,q = 1 σ(q) = p 0 otherwise, since the corresponding loss function is The squared Frobenius norm in this expression simply counts the number of mismatches between the observed but noisy permutations τ ji and the inferred permutations σ j σ −1 i .
Sentence 2: starting with the seminal work of Singer et al.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Most statistical regression models in use today are of the form: g(y) = f (x 1 )+f (x 2 )+• • •+f (x D ).
Sentence 2: the Dth order additive covariance function has D D = 1 term, a product of each dimension's covariance function: In the case where each base kernel is a one-dimensional squared-exponential kernel, the Dth-order term corresponds to the multivariate squared-exponential kernel: (5) also commonly known as the Gaussian kernel.
Relationship: neutral