sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We can try to more quantitatively describe the correctness of the inpainting procedure using as metrics the discriminator score and the leastsquares rigid-body alignment error with respect to the true native structure.
Sentence 2: the task of determining 3D cartesian coordinates given pairwise distance measurements is already well understood and has a natural formulation as a convex problem [31].
Relationship: neutral
Domain: NeurIPS
Sentence 1: In practice, an active-set strategy is introduced to further reduce cost of each iteration to the active size of variables and constraints for primal-sparse and dual-sparse LP respectively, where a primal-sparse LP has most of variables being zero, and a dual-sparse LP has few binding constraints at the optimal solution.
Sentence 2: the algorithm then conducts a projected line search to find smallest r ∈ {0, 1, 2, ...} satisfying and update x by x t,k+1 ← (x t,k + β r d * j ) + .
Relationship: neutral
Domain: NeurIPS
Sentence 1: In our proofs, we relied on the compactness of the set X .
Sentence 2: using Lemma (4), we get that Using similar arguments and using Equation (10) we can show that Summarizing, we have lim sup we can conclude that a.s. lim N →∞ To show that MHA is indeed a γ-bounded strategy, we use two special experts H 0,0 , H −1,−1 whose predictions are λ n 0,0 = λ max and λ n −1,−1 = 0 for every n and to shorten the notation, we denote g(y, λ, x) λ(c(y, x) − γ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Hence, the set of scale-space voxels covered is a 3D box, the width of the bounding box by its height (both discretized by a factor of 8 for efficiency) by 5, covering exactly one "octave" of scale space (i.e.
Sentence 2: we want to find points in feature space where the density of the positive set is large, and the density of the negative set is small.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The truncation level K c in the MGP is not fixed, but inferred from the adaptive sampler, with threshold parameter κ set to 0.05 (it is recommended to be set small for most applications).
Sentence 2: to motivate this model, consider the representation we may also express X = ΨΦ T , with Ψ ∈ R P ×Kc and Φ ∈ R N ×Kc ; the kth column of Ψ is defined by Lu :k and the kth column of Φ defined by Rv :k . Consequently, the low-rank assumption for H yields a low-rank model X = ΨΦ T , precisely as in [17,18].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Isomap (Figure 1(e)) performs poorly on this data set due to the low sampling rate on the manifold and the fact that the true mapping f is not isometric.
Sentence 2: if the true p Y follows the change of variable model (1), the recovered g converges to the true f −1 0 in the sense that they both describe a change of variable from the prior distribution p X to the distribution p Y .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The first can be bounded by the sum of the expected widths of the confidence intervals corresponding to each context x t i that might be chosen at each round t, where the expectation is over the randomness of the algorithm's distribution π t . A theorem of ?
Sentence 2: there are many settings in which the notion of merit we wish to predict may be different or even at odds with the notion of fairness we would like to enforce.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As we see, decreases sharply to less than 5 after all nonzero components have been identified.
Sentence 2: (3.9) the dictionary and its corresponding solution x B = x ⇤ B + xB and x N = 0 is optimal for the value of 2 [ ⇤ , max ], where Note that although initially the perturbations are nonnegative, as the dictionary gets updated, the perturbation does not necessarily maintain nonnegativity.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As an example of that, we here consider discriminative factor modeling, in which the factor scores are employed within a nonlinear SVM.
Sentence 2: in the E-step, we approximate the posterior of A, {Φ k }, ψ, f , λ and γ by a factorized distribution q(A) k q(Φ k )q(ψ)q(f )q(λ)q(γ) and in the M-step we optimize W and θ, using L-BFGS [23].
Relationship: neutral
Domain: NeurIPS
Sentence 1: There are a number of possible sign-function approximations, including hyperbolic and arc tangent, and shifted sigmoid function, the latter of which is the most flexible because of its free parameter β.
Sentence 2: under assumption B, an expression for the resulting α ij can be derived using the weak law of large numbers yielding: where x ∈ C i and x ∈ C j correspond to the samples as they are classified in a previous recognition pass; N i and N j are number of samples from model c i and c j respectively.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The bulk of an ERFNet is made up of standard residual blocks where each full convolution has been replaced by a pair of 1D alternatives: a 3 × 1 convolution followed by a 1 × 3 convolution.
Sentence 2: for example, in a Residual Network (ResNet) (He et al., 2016a) this standard block is a pair of sequential 3×3 convolutions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For a directed graph, the PageRank algorithm [19] aims to measure the nodes' importance by solving the linear problem , where d = 0.15 is called the "teleport" probability, N is the number of nodes and A is the column-normalized adjacency matrix.
Sentence 2: our strategy degrades gracefully as the number of stragglers increases, or as the deadline is pulled earlier.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Equivalence with the ECOC decision rule thus follows from the fact that 3 is the Hamming distance between the appropriate row of the coding matrix and the concatenation of the bits returned from the binary classifiers.
Sentence 2: if 7 © acts transitively on W then for all ¥ $ U¥ 21 ¢ W and I ¢ t . Proof.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In other words, we want to minimize H(z|s) and H(s) at every time frame.
Sentence 2: the sparsity parameter β was checked for various values from 0 to 0.8, and we used trained-basis models with 5, 10, 20, 40, 80, 160 and 320 bases, as well as the proposed scenario where all the training data is used as a dictionary.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Recall our assumption that the scope of each B C is only a small subset of variables S s . Then, the scope of R x is P S i pQ ( ¢¡ P £ ¥¤ §¦ P ¨ParentsP (% R Q .
Sentence 2: these value functions are a weighted linear combination of basis functions, as above, but each basis function is restricted to depend only on a small subset of state variables.
Relationship: neutral
Domain: NeurIPS
Sentence 1: From any initialization point γ (0) ∈ R m + the sequence of hyperparameter estimates {γ (k) } generated via γ (k+1) ∈ A(γ (k+1) ) is guaranteed to converge monotonically to a local minimum (or saddle point) of ( 2).
Sentence 2: at any non-minimizing γ the auxiliary cost function L z (γ) obtained from Step 3 will be strictly tangent to L(γ) at γ .
Relationship: neutral
Domain: NeurIPS
Sentence 1: A challenge for the interpretation of the principal components (or factors) is that they can be linear combinations of all the original variables.
Sentence 2: to get sparse principal components, we add a sparsity constraint to the optimization problem: every column of V should have at most r non-zero entries (the sparsity parameter r is an input), The sparse PCA problem is itself a very hard problem that is not only NP-hard, but also inapproximable [12] There are many heuristics for obtaining sparse factors [2,18,20,5,4,14,16] including some approximation algorithms with provable guarantees [1].
Relationship: neutral
Domain: NeurIPS
Sentence 1: As described in Section 2.3, the CorrLDA model first generates topics for the pivot language part of a document, and for the other language parts of the document, the model then uses the topics that were already generated in the pivot language.
Sentence 2: acknowledgments We thank Sinead Williamson, Manami Matsuura, and the anonymous reviewers for valuable discussions and comments.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This is because according to (10), when the primal-dual parameters (w, α) converge to the optimal parameters (w * , α * ), we have (∇φ i (w) + λnα i ) → 0, which means that even if the learning rate η t stays bounded away from zero, the procedure can converge.
Sentence 2: 3 shows more convex-case results (L2-regularized logistic regression) in terms of training loss residual (top) and test error rate (bottom) on rcv1.binary and covtype.binary from the LIBSVM site 2 , protein 3 , and CIFAR-10 4 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: We start by observing that Problem 5 really only depends on the integrals of monomials under the measure µ: for example, if , then we only need to know the integrals over the constituent monomials (y 3,0 : ) in order to evaluate the integral over f n . This suggests that we can optimize over the (parameter) moment sequence y = (y ↵ ) ↵2N P , rather than the measure µ itself.
Sentence 2: in typical polynomial optimization, we are only interested in solutions ✓ ⇤ that actually satisfy the given constraints, whereas here we are interested in K solutions , whose mixture satisfies constraints corresponding to the moment conditions (3).
Relationship: neutral
Domain: NeurIPS
Sentence 1: For instance, ENet authors rely on bottleneck residual blocks, sometimes employing dilation in its 3 × 3 convolution, or replacing it by two 1D convolutions (also possibly dilated).
Sentence 2: our approach learns the coordinates of the boxes in an end-to-end manner, and provides a generic layer that can be inserted into a ConvNet architecture multiple times.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Definition 1 Two sets A, A ∈ A are said to be neighbors if only a single hypothesis (and its complement, if it also belongs to H) outputs a different value on A and A .
Sentence 2: there exists a greedy procedure that yields query sequences that are within an O(log |H|) factor of the optimal search tree depth [GG74, KPB99, Lov85, AMM + 98, Das04], where |H| denotes the cardinality of H. The greedy procedure is referred to as Generalized Binary Search (GBS) [Das04,Now08] or the splitting algorithm [KPB99, Lov85, GG74]), and it reduces to classic binary search in special cases [Now08].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Formally, for any measurable function T : ⊗ k R n → {0, 1}, we have For k = 2, our result implies that, for β < 1, P 0 − P β TV is bounded away from 1.
Sentence 2: in this context we establish a lower bound on the critical signal-to-noise ratio below which a rank-one signal cannot be detected.
Relationship: neutral
Domain: NeurIPS
Sentence 1: At a high level BMM can be understood as an instance of the general assumed density filtering framework [14] where the algorithm finds an approximate posterior distribution within a tractable family of distributions by the method of moments.
Sentence 2: define w k,j α k,j / j α k,j , then where the second equality is by Corollary 1 that the network polynomial is a multilinear function of w k,j and the third equality holds because T F is the set of trees that do not contain w k,j .
Relationship: neutral
Domain: NeurIPS
Sentence 1: 3 In this section, we show that the policy gradient theorem does hold in the off-policy setting, when using function approximation for the policy, as long as we use emphatic weightings.
Sentence 2: we repeated the experiment in 5.1, with a chosen from {0, 0.25, 0.5, 0.75, 1} and the step-size chosen from {0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1}.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given a probability mass function P over X, x ∈ Val(X) is said to be a MAP instantiation for P if which means that x is the most likely value of X according to P . In principle a mass function P can have multiple (equally probable) MAP instantiations.
Sentence 2: figure 3 shows the classification accuracy for varying values of * that were used to perturb the model (in order to obtain the curves, the technicality was to split the test instances into bins according to the computed value * , using intervals of length 10 −2 , that is, accuracy was calculated for every instance with * between 0 and 0.01, then between 0.01 and 0.02, and so on).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our work distinguishes itself from these directions mainly in the objective, since we require the non-discrimination to happen in the long-term instead of at any time; this extends the classical batch definitions of non-discrimination in the online setting.
Sentence 2: the main representative of algorithms that do not have such a property are the algorithms that achieve low approximate regret compared to a shifting benchmark (tracking the best expert).
Relationship: neutral
Domain: NeurIPS
Sentence 1: coins of A, φ∼Uni(3MAJ n,∆n 1+µ ) (A(φ) = "typical") ≥ 1 − o(1) Assume first that φ ∈ 3MAJ n,∆n 1+µ is chosen at random.
Sentence 2: an n-variables 3MAJ formula is a boolean formula of the form where the C i 's are 3MAJ clauses.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The entries in the matrix Kt are given by that is the sum of the products of the weights over all paths of length t that start at vertex i and finish at vertex j in the weighted graph on the examples.
Sentence 2: such an approach suffers from well known limitations, mostly due to its inability to exploit semantic similarity between terms: documents sharing terms that are different but semantically related will be considered as unrelated.
Relationship: neutral
Domain: NeurIPS
Sentence 1: reparameterization) z = µ + R , where ∼ N (0, I dz ), we can generate samples from any Gaussian distribution N (µ, C) by simulating data from a standard normal distribution, provided the decomposition C = RR holds.
Sentence 2: each data point x follows a generative model p ψ (x|z), where this process is actually a decoder that is usually constructed by a non-linear transformation with unknown parameters ψ and a prior distribution p ψ (z).
Relationship: neutral
Domain: NeurIPS
Sentence 1: To the best of our knowledge, we are the first to study the identifiability and inference of GRUMs with multiple types.
Sentence 2: prior work in econometrics has focused on developing models that use data aggregated across types of agents, such as at the level of a geographic market, and that allow heterogeneity by using random coefficients on either agents' preference parameters [7,9] or on a set of dummy variables that define types of agents [6,27], or by imposing additional structure on the covariance matrix of idiosyncratic taste shocks [16].
Relationship: neutral
Domain: NeurIPS
Sentence 1: A naïve approach would be to simply reduce the dimensionality of the gradient vector, gt ∈ R τ = Πg t . ADA-FULL is now directly applicable in this low-dimensional space, returning a solution vector βt ∈ R τ at each iteration.
Sentence 2: we summarize the cost of each step in Algorithm 1 and contrast it with the cost of ADA-FULL in Table A.1 in Section A.
Relationship: neutral
Domain: NeurIPS
Sentence 1: (Note that we must do this inference without knowing the value of δ.)
Sentence 2: the probability that T δ will contain exactly one node in the path is Since, under this conditioning, the single node in the path will be uniformly distributed over the path itself, with half the probability it will be in the lower half of the path -causing the upper half of the path to be revealed.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given a possibly perturbed model M α , in which x ∼ N (0, Q α ), we have y ∼ N(0, S α ), where S α = rI + CQ α C T . Note that S α S α /r does not depend on r when is fixed, as Q α ∝ q ∝ r =⇒ S α ∝ r. Next, we aim to maximize the log-probability of y, which is a vector of N components (or pixels).
Sentence 2: 4e and f illustrate the procedure to find optimal q/r for a membrane and a plate, respectively, as explained in Section 2.3.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The mixture weights, reported in [33], are similar between the original scVI and our modification (and therefore higher than other mainstream differential expression procedures) and saturate the number of significant genes in this experiment (∼23%).
Sentence 2: the bias from the HSIC [17] is of order O( 1 n); it is negligible whenever the batch-size is large enough.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The algorithm recovers the hidden labels and alignments nearperfectly in this case.
Sentence 2: this expectation is then used in the M-step to calculate the re-estimation equations for updating the model parameters The posterior p(Φ i , z i |y i ) can be factorized as p zi (Φ|y i )p(z i |y i ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Repeating this process log 2 m times gives rise to a partition of the unit hypercube into m d hypercubes of identical size.
Sentence 2: finally, given {X j , Y j } i−1 j=1 the random variable X i does not depend in any way on f . Let fn : [0, 1] d → R denote an estimator based on the training samples {X i , Y i } n i=1 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Moreover, RMHMC suffers from expensive inner iteration.
Sentence 2: riemann manifold Langevin dynamics (RMLD) and Riemann manifold HMC (RMHMC) [13], are invented for better performance but still on the task of sampling in Euclidean space, where the target variable is treated as the global coordinates of some distribution manifold.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In practice, the estimated transition matrix has only one eigenvalue with unit magnitude and lim e→∞ ( ĀP ) e converges to a global stationary distribution.
Sentence 2: note that, in an actual implementation, one would use a low-rank reconstruction by zeroing-out the eigenvalues that are not equal to 1 in Λ to form ( ĀP ) r := V P ( ΛP ) r ( V P ) −1 = ( ĀP ) ∞ , where ( ΛP ) r ∈ R M K×M K is a diagonal matrix with only K non-zero entries, corresponding to the eigenvalues which are 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The probability distribution that x t = [x t σµ ] is expressed as where ⟨. . .⟩ A,W t ,X t denotes the average over the graph A, the random linear transform W t , and the state X t of the previous layer.
Sentence 2: the NMI is defined by We adopt this measure as the error function for the readout classifier.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this scheme the estimate will be based on fewer samples, but the samples that are used will be less correlated because they are more distant from each other in the Markov chain.
Sentence 2: 2 Algorithms 1 and 2 represent two of the three key ideas behind our proposal: both Gibbs sampling and Rao-Blackwellized estimation can be implemented efficiently using message passing on a junction tree.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We believe that these shortcomings, as well as the weak spot of RL models in not taking into account the internal drive state of the animal, all arise from lack of an integrated view toward learning and motivation.
Sentence 2: it has been argued by some authors [14] that even though habitual responses are insensitive to sensory-specific satiety, they are sensitive to motivational manipulations in a general, outcomeindependent way.
Relationship: neutral
Domain: NeurIPS
Sentence 1: More precisely, we shall consider a variational distribution that consists of a recognition model [4,2,10,8,15] which is parametrized by a "reverse" sigmoid network that predicts the latent vector x i from the associated observation y i : The variational parameters are contained in matrix V (also the bias terms).
Sentence 2: for such cases, reparametrization could be computationally more efficient than local expectation gradients since the latter approach will require to apply 1-D numerical integration to estimate the integral in (11) or the integral in ( 17) 2 which could be computationally more expensive.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Cats, many species of monkeys, and humans exibit ocular dominance stripes, which are alternating areas of primary visual cortex devoted to input from (the thalamic relay associated with) just one or the other eye (see Erwin et aI, 1995;Miller, 1996;Swindale, 1996 for reviews of theory and data).
Sentence 2: second, this value of C>w can be used to calculate the value of the normalisation parameter ).+ that affects the growth of 8W+ and 8W-.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We can see that for larger updates of the synapses (κ = 0.1) the network converges faster.
Sentence 2: we start approximating the terms N ǫ and (N − 1) ǫ by the sum of all pre-synaptic efficacies ǫ ik : This can be done for large N and if we suppose that the distribution of ǫ ik is the same for all i. E diss is now defined in terms of each individual neuron i as: An update of ǫ ij occurs when a spike from the pre-synaptic unit j induces a spike in a post-synaptic unit i.
Relationship: neutral
Domain: NeurIPS
Sentence 1: RSA revealed that the representational structures in the inferotemporal (IT) cortex of natural objects are highly similar between human and monkey [12] and a continuum in the abstract representation of biological classes exist in human ventral object visual cortex [2].
Sentence 2: if each single stimulus is modelled as a condition in X and regularization is used during regression, the correlation between β of temporally adjacent stimuli is higher primarily because of the autocorrelation property of the noise.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the limit, Λn → Λ Y = Λ X + Λ Z , a sum of a rank-k matrix and a diagonal matrix.
Sentence 2: we present an estimator which is appropriate when the conditional means E [y|u] lie in a low-dimensional linear space, and a maximum-likelihood estimator for additive Gaussian mixture noise.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As a result, instead of a single FV, describing the whole image, the image is represented by a large number of densely computed semi-local FVs, each of which describes a spatially adjacent set of local features, computed by the previous layer.
Sentence 2: during training, high-dimensional FVs, computed by the 2nd Fisher layer, are compressed using product quantisation [23].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Comparison of the optimal, true, generalised portraits to those based on the subject labels shows that classification has become more difficult: generalised portraits have moved closer to each other in gender space, narrowing the distance between the classes and thereby diminishing the gender typicality of the generalised portraits for all classifiers.
Sentence 2: if we disregard the constant offset b, we can write the decision function as x , which is a prototype classifier using the prototypes µ ± after whitening the space with S −1/2 W . We combine the linear preprocessor (PCA) X = EB and the linear classifier (SVM, RVM, Prot, FLD) y( x) = w| x + b to yield a linear classification system: y = w T E T + b where b = b 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Define: Then δ i is the amount of ε-insensitive error that can be corrected for example i by keeping all α ( * ) j =i constant and changing α ( * ) i . Optimality is reached by iteratively choosing the example with the largest δ i and changing its α ( * ) i within the [0, c] limits to minimize the error for this example.
Sentence 2: the form of the dual is max Note that the above expression conforms to the dual form of SVR.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The goal of the agent is to maximize its cumulative reward by acting optimally in the environment.
Sentence 2: compute the Q-values: Q( ✓S t , ✓a t ) = ( ✓a t ) T U ✓S t for a = 1, . . . , A t . 8: end for In this section, we develop an end-to-end learning algorithm for Q-LDA, by minimizing the loss function given in (7).
Relationship: neutral