Model: Master-thesis-NAP/ModernBERT-base-finetuned-hard_negatives
Author: Master-thesis-NAP
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, modernbert, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:79876, loss:MultipleNegativesRankingLoss, en, arxiv:1908.10084, arxiv:1705.00652, base_model:answerdotai/ModernBERT-base, base_model:finetune:answerdotai/ModernBERT-base, license:apache-2.0, model-index, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: answerdotai/ModernBERT-base
Card metadata: language: en
license: apache-2.0
tags: sentence-transformers, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:79876, loss:MultipleNegativesRankingLoss
base_model: answerdotai/ModernBERT-base
widget: source_sentence: What is the error estimate for the difference between the exact solution and the local oscillation decomposition (LOD) solution in terms of the $L_0$ norm?
sentences: [Werner-Young's inequality]\label{Young op-op}
Suppose $S\in \cS^p$ and $T\in \cS^q$ with $1+r^{-1}=p^{-1}+q^{-1}$.
Then $S\star T\in L^r(\R^{2d})$ and
\begin{align*}
    \|S\star T\|_{L^{r}}\leq \|S\|_{\cS^p}\|T\|_{\cS^q}.
\end{align*}, For the TT-ULV algorithm~\ref{alg:TT-ULV2} in the right-to-left sweep, if the error of the $k$th truncated ULV decomposition in step 4 is $\varepsilon_k$, then the total error of the computed TT-approximation tensor $\widehat{\mathcal{A}}$ satisfies
	\[ \| \mathcal{A}-\widehat{\mathcal{A}} \|_F \le \sum_{k=2}^{d} \varepsilon_k.\], \label{Corollary}
     Let Assumptions~\ref{assum_1} and~\ref{assump2} be satisfied. Let $u$ be the solution of~\eqref{WeakForm} and let $u_{H,k}$ be the LOD solution of~\eqref{local_probelm }. Then we have 
     \begin{equation}\label{L2Estimate}
         \|u-I_Hu_{H,k}\|_0\lesssim  \|u-I_Hu\|_0+\|u-u_{H,k}\|_0 +H|u-u_{H,k}|_1.
     \end{equation}
     %\[\|u-I_Hu_{H,k}\|_0\lesssim H |u|_1 +|u-u_{H,k}|_1.\], source_sentence: Does the theorem imply that the rate of convergence of the sequence $T_{m,j}(E)$ to $T_{m+k_n,j+k_n}(E)$ is exponential in the distance between $m$ and $j$, and that this rate is bounded by a constant $C$ times an exponential decay factor involving the parameter $\gamma$?
sentences: \label{theo:helper3}
Assume that $\{\PP_N\}_{N\ge 1}$ is a sequence of probability measures that is HT-appropriate in the sense of \cref{def:appropriate} and satisfies the LLN in the sense of \cref{def:LLN}.
Let $(\kappa_n)_{n\ge 1}$ and $(m_n)_{n\ge 1}$ be the sequences that arise from these definitions.
Moreover, assume that there exists a constant $C>0$ such that $|\kappa_n|\leq C^n$, for all $n \geq 1$.
Then $(m_n)_{n\ge 1}$ is the sequence of moments of a unique probability measure on $\R$., \label{lem1}
		For all $m,j\in\Z$,  we have
		\begin{equation*}
		|| T_{m,j} (E)-T_{m+k_n,j+k_n}(E)||\leq C e^{-\gamma  k_n}  e^{(\mathcal L(E)+\varepsilon) |m-j|}. 
		\end{equation*}, \label{RL1}
The system \eqref{R3} has the following positive fixed points if $0 <\alpha\leq1$ and $b>d$
$$E^*=\left(\dfrac{d}{b}, \dfrac{(b-d) r}{b^2}\right)$$, source_sentence: What is the expected value of the number of individuals in a Markov branching process with non-homogeneous Poisson immigration (MBPNPI) at time $t=0$, given that the immigration rate is $\lambda$?
sentences: The process $Y(t)$, $t\geq 0,$ is called Markov branching process with
non-homogeneous Poisson immigration (MBPNPI)., For any $\lambda \in(0,1)$ and $s \in\mathbb N$,
  \begin{equation*}
\sum_{k=s}^{\infty}\binom {k}{s}
(1-\lambda)^{k-s}=    \lambda^{-s-1}.
\end{equation*}, \label{lemma-sampling}
Fix an integer~$n\geq 1$.
Consider the initial configuration with one active particle on each
site of~$V_n$ and let the system evolve, with particles being killed
when they jump out of~$V_n$, until no active particle remains
in~$V_n$.
Then the distribution of the resulting stable configuration is exactly
the stationary distribution of the driven-dissipative Markov chain
on~$V_n$.
In particular, the number of sleeping particles remaining in~$V_n$ is
distributed as~$S_n$., source_sentence: What is the rate of convergence for the total error in the given problem, assuming the conditions in Theorem~\ref{convergence-rates} are met?
sentences: [Divergence Theorem or Gauss-Green Theorem for Surfaces in $\R^3$]
	\label{thm:surface_int}
	        Let $\Sigma \subset \Omega\subseteq\R^3$ be a bounded smooth surface.
	        Further, $\bb a:\Sigma\to\R^3$ is a continuously differentiable vector field that is either defined on the
					boundary $\partial\Sigma$ or has a bounded continuous extension to this boundary.
	        Like in \eqref{eq:decomp} it may be decomposed into tangential and normal components
					as follows $\bb a = \bb a^\shortparallel + a_\nu\bs\nu_\Sigma$. By $\dd l$ we denote the line element on 
					the curve $\partial \Sigma$. We assume that the curve is continuous and consists of finitely many
					smooth pieces.
	        Then the following divergence formula for surface integrals holds
	        %
	        \begin{align}
	            %
	            \int\limits_\Sigma \left[\nabla_\Sigma\cdot\bb a^\shortparallel\right](\x)\;\dd S
							= \int\limits_{\partial\Sigma} \left[\bb a\cdot\bs\nu_{\partial\Sigma}\right](\x)\,\dd l .
	            \label{eq:surface_div}
	            %
	        \end{align}
					%
					From this we obtain the formula
					%
	        \begin{align}
	            %
	            \int\limits_\Sigma \left[\nabla_\Sigma\cdot\bb a\right](\x)\;\dd S
							= \int\limits_{\partial\Sigma} \left[\bb a\cdot\bs\nu_{\partial\Sigma}\right](\x)\,\dd l 
							-\int\limits_\Sigma\left[ 2\kappa_Ma_\nu\right](\x)\;\dd S.
	            \label{eq:surface_div_2}
	            %
	        \end{align}
	    %, \label{convergence-rates}
Under the assumptions of Theorem~\ref{well-posedness}. Given $(\bu,{p},\bzeta,\varphi)\in (\bH^{s_1+1}(\Omega)\cap \bV_1)\times (\text{H}^{s_1}(\Omega)\cap Q_{b_1}) \times (\bH^{s_2}\cap \bV_2) \times (\text{H}^{s_2}\cap Q_{b_2})$, $(\bu_h,{p}_h,\bzeta_h,\varphi_h)\in \bV_1^{h,k_1}\times Q_1^{h,k_1}\times \bV_2^{h,k_2}\times Q_2^{h,k_2}$ be the respective solutions of the continuous and discrete problems, with the data satisfying $\fb\in \bH^{s_1-1}\cap \bQ_{b_1}$ and $g\in H^{s_2}(\Omega)\cap Q_{b_2}$. If $\overline{C}_1 \sqrt{M} L_\ell + \overline{C}_2^2 \sqrt{M^3} L_\bbM\sqrt{2\mu}   (\norm{\varphi_D}_{1/2,\Gamma_D} + \norm{g}_{0,\Omega}) < 1/2.$ Then, the total error $\overline{\textnormal{e}}_h:=\norm{(\bu-\bu_h,{p}-{p}_h, \bzeta-\bzeta_h,\varphi-\varphi_h)}_{\bV_1\times Q_{1} \times \bV_2\times Q_2}$ decays with the following rate for $s:= \min \left\{s_1,s_2\right\}$
    \begin{align*}\label{convergence-rate}
         \overline{\textnormal{e}}_h &\lesssim h^{ s} (|\fb|_{s_1-1,\bQ_{b_1}} + |\bu|_{s_1+1,\bV_1} + |{p}|_{s_1,Q_{b_1}} + |g|_{s_2,Q_{b_2}} + |\bzeta|_{s_2,\bV_2}+|\varphi|_{s_2,Q_{b_2}}).
    \end{align*}, [{\cite[Proposition~1]{Bychkov2021}}]\label{thm:bound_laurent}
  For every ODE system of the form~\eqref{eq:sys_main}, there exists a Laurent monomial quadratization with the order not exceeding the total number of terms in the right-hand side (that is, in $\bp$)., source_sentence: What is the error estimate for the eigenfunction approximation in terms of the weak eigenvalue and the norm of the difference between the exact and approximate eigenfunctions?
sentences: \big[{\bf Condition $SD1(h)$}\big]\label{DefnSD1(h)}

In \cite{MDL} an approximation order $O(h^s)$, as $h\to 0$, is proved, where $h$ is the sampling distance. The achievable order $s$ is of course limited by the smoothness order of the boundaries of $Graph(F)$. Then, the order $s$ depends upon the degree of the polynomials used to approximate the boundary near the neighborhood of points of topology change and upon the degree of splines used at regular regions. 

For example, let us view Step C of the approximation algorithm described in Section 5.2 of \cite{MDL}. 
It is assumed that the boundary curves are $C^{2k}$ smooth, and it is implicitly assumed that $h$ is small enough so that there are $2k$ sample points close to the point of topology change, for computing the polynomial $p_{2k-1}$ therein.
This condition is related to the more general condition $SD(h)$ and it can serve as a practical way of checking it for the case $d=1$. That is, near a point of topology change, we check whether there are enough sample points for applying the approximation algorithm in \cite{MDL}. We denote this condition as the $SD1(h)$ condition., Consider dynamics \eqref{avg} and define the corresponding average dynamics as $\label{T-avg}
\mathring{\chi} = \epsilon h_{av}(\chi)$, with the average function defined as
\begin{equation*} 
h_{av}(\chi):=\lim_{T \to \infty} \frac{1}{T}\int_{t}^{t+T} h(\mu, \chi, 0) d \mu, \ T>0,
\end{equation*}
both \eqref{avg} and \eqref{T-avg} twice differentiable and bounded in every compact set of the $\chi$-domain $\mathcal{D} \subset \mathbb{R}^{3}$. 
%
Let $\chi(\tau,\epsilon)$ and $\chi_{av}(\epsilon\tau)$ denote the solutions of \eqref{avg} and \eqref{T-avg}, respectively. If $\chi_{av}(\epsilon\tau)\in \mathcal{D}$ for all $\tau\in[0,\zeta/\epsilon]$, $\zeta\geq 0$, and $\chi(0,\epsilon) - \chi_{av}(0)=\mathcal{O}(\nu(\epsilon))$, then there exists an $\epsilon^{*}>0$ such that for all $0<\epsilon<\epsilon^{*}$, $\chi(\tau,\epsilon)$ is well defined and
$$
\chi(\tau,\epsilon) - \chi_{av}(\epsilon\tau) = \mathcal{O}(\nu(\epsilon)) \ \textnormal{on} \ \tau \in [0, \zeta/\epsilon],
$$
for some function $\nu\in \mathcal{K}$., (\cite{DangWangXieZhou})\label{Theorem_Error_Estimate_k}
Let us define the spectral projection $F_{k,h}^{(\ell)}: V\mapsto {\rm span}\{u_{1,h}^{(\ell)}, \cdots, u_{k,h}^{(\ell)}\}$ for any integer $\ell \geq 1$ as follows:
\begin{eqnarray*}
a(F_{k,h}^{(\ell)}w, u_{i,h}^{(\ell)}) = a(w, u_{i,h}^{(\ell)}), \ \ \ i=1, \cdots, k\ \ {\rm for}\ w\in V.
\end{eqnarray*}
Then the exact eigenfunctions $\bar u_{1,h},\cdots, \bar u_{k,h}$ of (\ref{Weak_Eigenvalue_Discrete}) and the eigenfunction approximations $u_{1,h}^{(\ell+1)}$, $\cdots$,  $u_{k,h}^{(\ell+1)}$ from Algorithm \ref{Algorithm_k} with the integer $\ell > 1$ have the following error estimate:
\begin{eqnarray*}\label{Error_Estimate_Inverse}
 \left\|\bar u_{i,h} - F_{k,h}^{(\ell+1)}\bar u_{i,h} \right\|_a \leq
 \bar\lambda_{i,h} \sqrt{1+\frac{\eta_a^2(V_H)}{\bar\lambda_{1,h}\big(\delta_{k,i,h}^{(\ell+1)}\big)^2}}
\left(1+\frac{\bar\mu_{1,h}}{\delta_{k,i,h}^{(\ell)}}\right)\eta_a^2(V_H)\left\|\bar u_{i,h} - F_{k,h}^{(\ell)}\bar u_{i,h} \right\|_a,
\end{eqnarray*}
where $\delta_{k,i,h}^{(\ell)} $ is defined as follows:
\begin{eqnarray*}
\delta_{k,i,h}^{(\ell)} = \min_{j\not\in \{1, \cdots, k\}}\left|\frac{1}{\lambda_{j,h}^{(\ell)}}-\frac{1}{\bar\lambda_{i,h}}\right|,\ \ \ i=1, \cdots, k.
\end{eqnarray*}
Furthermore, the following $\left\|\cdot\right\|_b$-norm error estimate holds:
\begin{eqnarray*}
\left\|\bar u_{i,h} -F_{k,h}^{(\ell+1)}\bar u_{i,h} \right\|_b\leq 
\left(1+\frac{\bar\mu_{1,h}}{\delta_{k,i,h}^{(\ell+1)}}\right)\eta_a(V_H) \left\|\bar u_{i,h} -F_{k,h}^{(\ell+1)}\bar u_{i,h}\right\|_a.
\end{eqnarray*}
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics: cosine_accuracy@1, cosine_accuracy@3, cosine_accuracy@5, cosine_accuracy@10, cosine_precision@1, cosine_precision@3, cosine_precision@5, cosine_precision@10, cosine_recall@1, cosine_recall@3, cosine_recall@5, cosine_recall@10, cosine_ndcg@10, cosine_mrr@10, cosine_map@100
model-index: name: ModernBERT base
results: task: type: information-retrieval
name: Information Retrieval
dataset: name: TESTING
type: TESTING
metrics: type: cosine_accuracy@1
value: 0.8599446239040147
name: Cosine Accuracy@1
verified: no, type: cosine_accuracy@3
value: 0.9112828795569913
name: Cosine Accuracy@3
verified: no, type: cosine_accuracy@5
value: 0.9272035071527457
name: Cosine Accuracy@5
verified: no, type: cosine_accuracy@10
value: 0.9432395016151361
name: Cosine Accuracy@10
verified: no, type: cosine_precision@1
value: 0.8599446239040147
name: Cosine Precision@1
verified: no, type: cosine_precision@3
value: 0.6011767420396862
name: Cosine Precision@3
verified: no, type: cosine_precision@5
value: 0.4808721735117674
name: Cosine Precision@5
verified: no, type: cosine_precision@10
value: 0.3352445777572682
name: Cosine Precision@10
verified: no, type: cosine_recall@1
value: 0.04145783702387875
name: Cosine Recall@1
verified: no, type: cosine_recall@3
value: 0.08131110706944317
name: Cosine Recall@3
verified: no, type: cosine_recall@5
value: 0.1041094567622933
name: Cosine Recall@5
verified: no, type: cosine_recall@10
value: 0.1366201031745754
name: Cosine Recall@10
verified: no, type: cosine_ndcg@10
value: 0.43733031244545484
name: Cosine Ndcg@10
verified: no, type: cosine_mrr@10
value: 0.8887459621596662
name: Cosine Mrr@10
verified: no, type: cosine_map@100
value: 0.1551529611644354
name: Cosine Map@100
verified: no
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Created: 2025-05-28 11:42:43