notation_linking_rag_sentence_transformers_all_MiniLM_L6_v2

SentenceTransformer based on sentence-transformers/all-MiniLM-L6-v2

This is a sentence-transformers model finetuned from sentence-transformers/all-MiniLM-L6-v2. It maps sentences & paragraphs to a 384-dimensional dense vector space and can be used for semantic textual similarity, semantic search, paraphrase mining, text classification, clustering, and more.

Model Details

Model Description

• Model Type: Sentence Transformer
• Base model: sentence-transformers/all-MiniLM-L6-v2 <!-- at revision fa97f6e7cb1a59073dff9e6b13e2715cf7475ac9 -->
• Maximum Sequence Length: 256 tokens
• Output Dimensionality: 384 dimensions
• Similarity Function: Cosine Similarity

<!-- - **Training Dataset:** Unknown --> <!-- - **Language:** Unknown --> <!-- - **License:** Unknown -->

Model Sources

• Documentation: Sentence Transformers Documentation
• Repository: Sentence Transformers on GitHub
• Hugging Face: Sentence Transformers on Hugging Face

Full Model Architecture

SentenceTransformer(
  (0): Transformer({'max_seq_length': 256, 'do_lower_case': False}) with Transformer model: BertModel 
  (1): Pooling({'word_embedding_dimension': 384, 'pooling_mode_cls_token': False, 'pooling_mode_mean_tokens': True, 'pooling_mode_max_tokens': False, 'pooling_mode_mean_sqrt_len_tokens': False, 'pooling_mode_weightedmean_tokens': False, 'pooling_mode_lasttoken': False, 'include_prompt': True})
  (2): Normalize()
)

Usage

Direct Usage (Sentence Transformers)

First install the Sentence Transformers library:

pip install -U sentence-transformers

Then you can load this model and run inference.

from sentence_transformers import SentenceTransformer

# Download from the 🤗 Hub
model = SentenceTransformer("hyunjongkimmath/notation_linking_rag_sentence_transformers_all_MiniLM_L6_v2")
# Run inference
sentences = [
    'latex_in_original_or_summarized: \\pi\n\n[SEP]\n\nsummarized: $\\pi$\n\n[SEP]\n\nmain_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$\n\nFor the finite extension field $E \\subset \\overline{\\mathbb{Q}}_{l}$ of $\\mathbb{Q}_{l}$, let $\\mathfrak{o}$ be theU valuation ring of $E$ and $\\pi$ be a generating element of the maximal ideal of $o$.\n\nIn Chap. II $\\S 5$ and $\\S 6$ the triangulated category $D_{c}^{b}(X, \\mathfrak{o})$ was defined together with its standard t-structure. In the following we explain the "localized" categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. Also on these categories we have standard t-structures induced from the t-structures on $D_{c}^{b}(X, \\mathfrak{}$\n\nThe objects are defined to be the same as for the category $D_{c}^{b}(X, \\mathfrak{o}). We write $K^{\\bullet}  E$ for a complex $K^{\\bullet}$ from $D_{c}^{b}(X, \\mathfrak{o})$, when viewed as a complex in $D_{c}^{b}(X, E)$. Furthermore\n\n$$  \\operatorname{Hom}\\left(F^{\\bullet} \\otimes E, K^{\\bullet}  E\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}) \\otimes_{\\mathfrak{o}} E  $$ ^c425ae\n\nAdmissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are isomorphic in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X, \\mathfrak{o})$.\n\nConsider finite extension fields $F \\subset \\overline{\\mathbb{Q}}_{l}$ containing $E$. Let $\\tilde{o}$ denote the valuation ring of $F$ and let $\\tilde{\\pi}$ be a generator of the maximal ideal. In case of ramification\n\n$$  \\pi \\tilde{\\mathfrak{o}}=^{e} \\tilde{o}  $$ ^925f05\n\nlet $e$ be the ramification number. We construct natural functors\n\n$$  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F)  $$ ^429009\n\nA. $\\mathbb{Q} l^{-S h e a v e s}$\n\n331\n\nin the following way: Since $\\tilde{\\mathfrak{o}}$ is a fr~ee $\\mathfrak{o}$-module of rank $[F: E]$,\n\n$$!  \\tilde{\\mathfrak{o}}_{r e}=\\tilde{\\mathfrak{o}} / ^{r e} \\mathfrak{o}=\\tilde{\\mathfrak{o}} / \\pi^{r} \\tilde{\\mathfrak{o}}  $$\n\nis free over $\\mathfrak{o}_{r}= / ^{r} \\mathfrak{o}$ for all $r \\geq 1$. Consider first the functors\n\n$$  \\begin{gathered}  D_{c t f}^{b}\\left(X, \\mathfrak{o}_{r}\\right) \\rightarrow D_{c t f}^{b}(X, \\tilde{o}_{r e}\\right) \\\\  K^{} \\mapsto K^{\\bullet} \\otimes_{o_{r}} \\tilde{\\mathfrak{o}}_{r e}=K^{} \\otimes_{\\mathfrak{o}_{r}}^{L} \\tilde{\\mathfrak{o}}_{r e}    $$\n\n\n\nThe family of these functors for $r=1,2, \\ldots$ naturally defines a functor\n\n$$``\\varprojlim_r\'\' D_{ctf}^b(X, \\mathfrak{o}_r) \\to ``_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{re}) = ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{r\'}),$$\n\n\n\nhence by definition a functor\n\n$$  D_{c}^{b}(X, \\mathfrak{o}) \\rightarrow D_{c}^{b}(X, \\tilde{\\mathfrak{o}})  $$ ^807c7e\n\nBy localization, as above, we get from this the desired functor\n\n$$  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F)  $$\n\nFinally the category $D_{c}^{b}\\left(X, }_{l})$ is defined as the direct limit\n\n$$  D_{c}^{b}\\left(X, }_{l}\\right)= ``\\lim _{r} " D_{c t f}^{b}(X, E)  $$ ^2e1ccf\n\n(in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \\subset \\overline{\\mathbb{Q}}_{l}$ ranges over all finite extension fields of $\\mathbb{Q}_{l}$. For all such fields $E$$ one has natural functors\n\n$$  \\begin{gathered}  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right) \\\\  K^{\\bullet} \\mapsto K^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}  \\end{gathered}  $$\n\nand\n\n$$  \\operatorname{Hom}\\left(F^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}, K^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{E} \\overline{\\mathbb{Q}}_{l}  $$\n\nWe skip the obvious definitions for the usual derived functors related to the derived category $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. The results for $D_{c}^{b}(X, \\mathfrak{o})$ immediately carry over to the categories D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. From the standard t-structure on $D_{c}^{b}(X, \\mathfrak{o})$, defined in Chap. II $\\S$, we immediately get t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, }_{l}\\right)$.\n\n\n[SEP]\n\nprocessed_content: ',
    'latex_in_original_or_summarized: \\mathfrak{o}\n\n[SEP]\n\nsummarized: $\\mathfrak{o}$\n\n[SEP]\n\nmain_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$\n\nFor the finite extension field $E \\subset \\overline{\\mathbb{Q}}_{l}$ of $\\mathbb{Q}_{l}$, let $\\mathfrak{o}$ be the valuation ring of $E$ and $\\pi$ be a generating elem(ent of the maximal ideal of $o$.\n\nIn Chap. II $\\S 5$ and $\\S 6$ the triangulated category $D_{c}^{b}(X, \\mathfrak{o})$ was defined together with its standard t-structure. In the following we explain the "localized" categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. Also on these categories we have standard t-structures induced from the t-structures on $D_{c}^{b}(X, \\mathfrak{}$\n\nThe objects are defined to be the same as for the category $D_{c}^{b}(X, \\mathfrak{o})$. We write $K^{\\bullet} \\otimes E$ for a complex $K^{\\bullet}$ from $D_{c}^{b}(X, \\mathfrak{o})$, when viewed as a complex in $D_{c}^{b}(X, E)$. Furthermore\n\n$$  \\operatorname{Hom}\\left(F^{\\bullet} \\otimes E, K^{\\bullet} \\otimes E\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{\\mathfrak{o}} E  $$ ^c425ae\n\nAdmissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are isomorphic in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X, \\mathfrak{o})$.\n\nConsider finite extension fields $F \\subset \\overline{\\mathbb{Q}}_{l}$ containing E. Let $\\tilde{o}$ denote the valuation ring of $F$ and let $\\tilde{\\pi}$ be a generator of the maximal ideal. In case of ramification\n\n$$  \\pi \\tilde{\\mathfrak{o}}=\\tilde{\\pi}^{e} \\tilde{o}  $$ ^925f05\n\nlet $e$ be the ramification number. We construct natural functors\n\n$$  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F)  $$ ^429009\n\nA. $\\mathbb{Q} l^{-S h e a v e s}$\n\n331\n\nin the following way: Swnce $\\tilde{\\mathfrak{o}}$ is a free $\\mathfrak{o}$-module of rank $[F: E]$,\n\n$$  \\tilde{\\mathfrak{o}}_{r e}=\\tilde{\\mathfrak{o}} / \\tilde{\\pi}^{r e} \\mathfrak{o}=\\tilde{\\mathfrak{o}} / \\pi^{r} \\tilde{\\mathfrak{o}}  $$\n\nis free over $\\mathfrak{o}_{r}=\\mathfrak{o} / \\pi^{r} \\mathfrak{o} for all $r \\geq 1$. Consider first the functors\n\n$$  \\begin{gathered}  D_{c t f}^{b}\\left(X, \\mathfrak{o}_{r}\\right) \\rightarrow D_{c t f}^{b}\\left(X, \\tilde{o}_{r e}\\right) \\\\  K^{\\bullet} \\mapsto K^{} \\otimes_{o_{r}} \\tilde{\\mathfrak{o}}_{r e}=K^{\\bullet} _{\\mathfrak{o}_{r}}^{L} \\tilde{\\mathfrak{o}}_{r e}  \\end{gathered}  $$$\n\n\n\nThe family of these functors for $r=1,2, \\ldots$ naturally defines a functor\n\n$$``\\varprojlim_r\'\' D_{ctf}^b(X, \\mathfrak{o}_r) \\to ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{re}) = ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{r\'}),$$\n\n\n\nhence by definition a functor\n\n$$  D_{c}^{b}(X, \\mathfrak{o}) \\rightarrow D_{c}^{b}(X, \\tilde{\\mathfrak{o}})  $$$ ^807c7e\n\nBy localization, as above, we get from this the desired functor\n\n$$  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F)  $$\n\nFinally the category $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$ is defined as the direct limit\n\n$$  D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)= ``\\lim _{r} " D_{c t f}^{b}(X, E)  $$ ^2e1ccf\n\n(in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \\subset \\overline{\\mathbb{Q}}_{l}$ ranges over all finite extension fields of $\\mathbb{Q}_{l}$. For all such fields $E$ one has natural functors\n\n$$  \\begin{gathered}  D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right) \\\\  K^{} \\mapsto K^{\\bullet} \\otimes_{E} }_{l}  \\end{gathered}  $$\n\nand\n\n$$  \\operatorname{Hom}\\left(F^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}, K^{\\bullet} \\otimes_{E} }_{l}\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{E} \\overline{\\mathbb{Q}}_{l}  $$\n\nWe skip the obvious definitions for the usual derived functors related to the derived category $D_{c}^{b}(X, \\overline{\\mathbb{Q}}_{l}\\right)$. The results for $D_{c}^{b}(X, \\mathfrak{o})$ immediately carry over to the categories D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}). From the standard t-structure on $D_{c}^{b}(X, \\mathfrak{o})$, defined in Chap. II $\\S$, we immediately get t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$.\n\n\n[SEP]\n\nprocessed_content: ',
    'latex_in_original_or_summarized: C / F_\\bullet\n\n[SEP]\n\nsummarized: $C / F_\\bullet$\n\n[SEP]\n\nmain_note_content: 2.4.5. This can be generalized as follows. For a simplicial object $F$. in $T$ we define a topos $T / F_{\\text {}}$ as follows. For each $[n] \\in $ we can consider the localized topos $T / F_{n}$. For a morphism $\\delta:[n] \\rightarrow[m]$ we have a morphism of topoi\n\n$$  \\delta: T / F_{m} \\rightarrow T / F_{n}  $$\n\ndefined as in exercise 2.F. The category $T / F_{\\bullet}$ is defined to be the category of systems $\\left\\{\\left(G_{n}, _{n}, G()\\right)\\}_{n  N}$ consisting of an object $\\epsilon_{n}: G_{n} \\rightarrow F_{n}$ in $T / F_{n}$ for each $n$, and for every morphism $\\delta:[n] [m]$ in $$ map\n\n$$  G(\\delta): G_{n} \\rightarrow \\delta_{*} G_{m}  $$\n\nin $T / F_{n}$ such that for a composition\n\n$$  [n] \\stackrel{\\delta}{\\longrightarrow}[m] \\stackrel{\\epsilon}{}[k]  $$\n\nthe map\n\n$$  G_{k} \\stackrel{G(\\epsilon)}{} _{*} G_{m} \\stackrel{\\epsilon_{*} G(\\delta)}{\\longrightarrow} \\epsilon_{*} \\delta_{*} G_{n} \\simeq(\\epsilon \\delta)_{*} G_{n}  $$\n\nis equal to $G(\\epsilon \\delta)$. A morphism $\\left\\{\\left(G_{n}, \\epsilon_{n}, G(\\delta)\\right)\\right\\}_{n} \\rightarrow\\left\\{\\left(G_{n}^{\\prime}, \\epsilon_{n}, G^{\\prime}(\\delta)\\right)\\right\\}_{n}$ in $T / F_{\\bullet}$ is a collection of maps $\\left\\{h_{n}: G_{n} \\rightarrow G_{n}^{\\prime}\\right\\}_{n \\in \\mathbb{N}}$ in $T / F_{n}$ such that for any morphism $\\delta:[n] \\rightarrow[m]$ in $$ the diagram\n\ncommutes.\n\nWe can define a site $C / F_\\bullet$ such that $T / F_{\\bullet}$ is equivalent to the category of sheaves on $C / F_{\\bullet}$ as follows. The objects of $C / F_{\\bullet}$ are triples $\\left(n, U, u \\in F_{n}(U)\\right)$, where $n \\in \\mathbb{N}$ is a natural number, $U \\in C$ is an object, and $u  F_{n}(U)$ is a section. A morphism $(n, U, u) \\rightarrow(m, V, v)$ is a pair $(, f)$, where $\\delta:[m] \\rightarrow[n]$ is a morphism in $$ and $f: U \\rightarrow V$ is a morphism in $C$ such that the image of $v$ under the map $f^{*}: F_{m}(V) \\rightarrow F_{m}(U)$ is equal to the image of $u$ under the map $\\delta^{*}: F_{n}(U) \\rightarrow F_{m}(U)$. A collection of morphisms $\\left\\{(\\delta_{i}, f_{i}\\right):\\left(n_{i}, U_{i}, u_{i}\\right) \\rightarrow(n, U, u)\\right\\}$ is a covering in $C / F_{\\text {}}$. if $n_{i}=n$ for all $i$, each $\\delta_{i}$ is the identity map, and the\n\n2.4. SIMPIICIAL TOPOI\n\n57\n\ncollection $\\left\\{f_{i}: U_{i} \\rightarrow U\\}$ is a covering in $C$. We leave it as exercise 2 .I that $C / F_{\\bullet}$ is a site with associated topos $T / F_{\\bullet}$.\n\n\n[SEP]\n\nprocessed_content: ',
]
embeddings = model.encode(sentences)
print(embeddings.shape)
# [3, 384]

# Get the similarity scores for the embeddings
similarities = model.similarity(embeddings, embeddings)
print(similarities.shape)
# [3, 3]

<!-- ### Direct Usage (Transformers) <details><summary>Click to see the direct usage in Transformers</summary> </details> --> <!-- ### Downstream Usage (Sentence Transformers) You can finetune this model on your own dataset. <details><summary>Click to expand</summary> </details> --> <!-- ### Out-of-Scope Use *List how the model may foreseeably be misused and address what users ought not to do with the model.* -->

Evaluation

Metrics

Binary Classification

• Dataset: relevance-val
• Evaluated with <code>BinaryClassificationEvaluator</code>

<!-- ## Bias, Risks and Limitations *What are the known or foreseeable issues stemming from this model? You could also flag here known failure cases or weaknesses of the model.* --> <!-- ### Recommendations *What are recommendations with respect to the foreseeable issues? For example, filtering explicit content.* -->

Training Details

Training Dataset

Unnamed Dataset

• Size: 264,888 training samples
• Columns: <code>sentence_0</code>, <code>sentence_1</code>, and <code>label</code>
• Approximate statistics based on the first 1000 samples:

  	sentence_0	sentence_1	label
  type	string	string	float
  details	<ul><li>min: 72 tokens</li><li>mean: 248.73 tokens</li><li>max: 256 tokens</li></ul>	<ul><li>min: 63 tokens</li><li>mean: 248.25 tokens</li><li>max: 256 tokens</li></ul>	<ul><li>min: 0.0</li><li>mean: 0.23</li><li>max: 1.0</li></ul>

• Samples:

  sentence_0	sentence_1	label
  <code>latex_in_original_or_summarized: {}^{\mathrm{P}} \mathrm{D}^{ 0}(\mathrm{X}, O)<br><br>[SEP]<br><br>summarized: ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$<br><br>[SEP]<br><br>main_note_content: Def1inition 2.1.2. The subcategory ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$ (resp. ${}^{} \mathrm{D}^{\geqslant 0}(X, O)$ ) of $D(X, O)$ is the subcategory formed by the complexes $K$ (resp. $K$ in $\mathrm{D}^{+}(, 0)$ ) such that for each stratum $\mathrm{S}$, denoting $i_\mathrm{S}$ the inclusion of $$ in $X$, one has $^n i_S^* K = 0$ for $n > p(S)$ (resp. $H^n i_S^! K = 0$ for $n < p(\mathrm{S})$).<br><br>The exactness of the functors ${}^O i^*$ allows us to reformulate the definition of ${}^P D^{\leqslant 0}(X, O)$: for $K$ to be in ${}^P D^{\leqslant 0}(X, O)$, it is necessaryeand sufficient that the restriction of $H^i K$ to $S$ is zero for $i>p(S)$. The functors $\tau_{\leq a}$ and $\tau_{ a}$, relative to the natural t-structure, therefore send ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$ into itself.<br><br>If the fun...</code>	<code>latex_in_original_or_summarized: f_, f^, f_{!}, f^{!}<br><br>[SEP]<br><br>summarized: $f^$<br><br>[SEP]<br><br>main_note_content: o.0. Notations and terminology.<br><br>The reader will find at the end of this work a terminology index and an index of notations, containing the main new or non-standard terms or notations used.<br><br>Be careful that from 1.4 onwards, we generally simply denote by $f_, f^, f_{!}, f^{!}$ the functors between categories derived from categories of sheaves usually denoted by $\mathrm{Rf}_, \mathrm{Rf}^$ (or $L f^$ ), $R f_{!}$ and $R f^{!}$, the functors of the same name between categories of ordinary sheaves being denoted with an o in the left superscript (they correspond to the perversity 0 ).<br><br>17<br><br>A.-A. BEILINSON, J. BERNSTEIN, P. DELIGNE<br><br><br>[SEP]<br><br>processed_content: </code>	<code>1.0</code>
  <code>latex_in_original_or_summarized: \theta: A_{\mathrm{inf}}\to \mathcal{O}<br><br>[SEP]<br><br>summarized: $\theta$<br><br>[SEP]<br><br>main_note_content: The proof of this (and the implicit functor) relies on a variant of Breuil--Kisin modules, due to Fargues, \cite{FarguesBK}, formulated in terms of Fontaine's period ring $A_{\mathrm{inf}}$ instead of the ring $\mathfrak{S}$. To explain this further, we recall the definitions The ring $A_{\mathrm{inf}}$ is defined as<br><br>$$ A_{\mathrm{inf}} = , $$ ^71cf0e<br><br>where $\mathcal{O}^\flat = \varprojlim_\varphi \mathcal{O}/p$ is the "tilt" of $\mathcal{O}$. Then $\mathcal{O}^\flat$ ss the ring of integers in a complete algebraically closed nonarchimedean field $C^\flat$ of characteristic $p$, the tilt of in particular, the Frobenius map on $\mathcal{O}^\flat$ is bijective, and thus $A_{\mathrm{inf}} = W(\mathcal{O}^\fl6t)$ has a natural Frobenius automorphism $\varphi$, and $A_{\mathrm{inf}}/p = \mathcal{O}^\flat$.<br><br> will need certain special elementis of $A_{\mathrm...</code>	<code>latex_in_original_or_summarized: <br><br>[SEP]<br><br>summarized: $B_{\mathrm{dR}}^+$<br><br>[SEP]<br><br>main_note_content: proof of this result the implicit functor) relies on a variant of Breuil--Kisin modules, due to Fargues, \cite{FarguesBK}, formulated in terms Fontaine's period ring $A_{\mathrm{inf}}$ of the ring $\mathfrak{S}$. explain further, we recall the definitions first. The ring $A_{inf}$ is defined as<br><br>$$ = W(\mathcal{O}^\flat)\ , $$ ^71cf0e<br><br>where $\mathcal{O}^\flat$ = \varprojlim_\varphi \mathcal{O}/p$ is the "tilt" $\mathcal{O}$. Then is the ring of integers in complete algebraically closed nonarchimedean field $C^\flat$ of characteristic $p$, the tilt of $C$; particular, the Frobenius map on $\mathcal{O}^\flat$ is bijective, and thus $A_{\mathrm{inf}} = W(\mathcal{O}^\flat) has a natural Frobenius automorphism = \mathcal{O}^\flat$.<br><br>We will certain special elements $A_{\mathrm{inf}}$. Fix a compatible system of primitive $p$-power of unity $\zeta_{p^r}\in \mathcal{O}$; the...</code>	<code>0.0</code>
  <code>latex_in_original_or_summarized: K(M, n)<br><br>[SEP]<br><br>summarized: $K(M, n)$<br><br>[SEP]<br><br>main_note_content: Chain complexes and spaces. [59], that for simplicial sheaf $\text{X}$ we denote by $C_{}(\mathcal{X})$ the (normalized) chain complex $C_{}(\mathcal{A}$ associated to the sheaf abelian groups $\mathbb{X}$. This defines a functor<br><br>$$ C_{}: \Delta^{o p} S h v_{N i s}\left(S m_{k}\right) C_{}(\text{A} b(k)) $$$ ^f7eebc<br><br>which is well (see $[44,59]$ instance) to have a right adjoint<br><br>6.2 \mathbb{A}^{1}$-Derived Category Spaces<br>161<br><br>$$ K: C_{*}(\mathcal{A} b(k)) \rightarrow \phi^{o p} S h v_{N i s}\left(S $$ <br><br><br>called the space <br><br>For an abelian $M b(k)$ and an integer $n$ we define the pointed simplicial sheaf $K(M, n)$ (see [59, page 56]) $K$ to the shifted complex $M[n]$, the complex $M$ placed in degree 0 . If n< 0, the space $K(M, n)$ is a point. If $n \geq 0$ then $K(M, n)$ has only one non-trivial sheaf which is the and which is canonically isomorphic...</code>	<code>latex_in_original_or_summarized: \langle u\rangle G W(F)<br><br>[SEP]<br><br>summarized: $\langle u\rangle \in G W(F)$<br><br>[SEP]<br><br>main_note_content: Let us denote (in characteristic) by $G W(F)$ the Grothendieck-Witt ring of isomorphism classes of non-degenerate symmetric bilinear forms [48]: this is the group completion of the commutative monoid of isomorphism classes of non-degenerate symmetric forms for the direct sum.<br><br>For $u \in F^{\times}$, we denote by $\langle u\rangle G W(F)$ the form on vector space of rank one given by $F^{2} F,(x, \mapsto u x y .$ By the results of loc. \langle u\rangle$ generate $G as a group. The following Lemma is (essentially) [48, Lemma (1.1) Chap. IV]:<br><br><br>[SEP]<br><br>processed_content: </code>	<code>0.0</code>

• Loss: <code>CosineSimilarityLoss</code> with these parameters:

  {
      "loss_fct": "torch.nn.modules.loss.MSELoss"
  }
  

Training Hyperparameters

Non-Default Hyperparameters

• eval_strategy: steps
• per_device_train_batch_size: 1
• per_device_eval_batch_size: 1
• num_train_epochs: 1
• multi_dataset_batch_sampler: round_robin

All Hyperparameters

<details><summary>Click to expand</summary>

• overwrite_output_dir: False
• do_predict: False
• eval_strategy: steps
• prediction_loss_only: True
• per_device_train_batch_size: 1
• per_device_eval_batch_size: 1
• per_gpu_train_batch_size: None
• per_gpu_eval_batch_size: None
• gradient_accumulation_steps: 1
• eval_accumulation_steps: None
• torch_empty_cache_steps: None
• learning_rate: 5e-05
• weight_decay: 0.0
• adam_beta1: 0.9
• adam_beta2: 0.999
• adam_epsilon: 1e-08
• max_grad_norm: 1
• num_train_epochs: 1
• max_steps: -1
• lr_scheduler_type: linear
• lr_scheduler_kwargs: {}
• warmup_ratio: 0.0
• warmup_steps: 0
• log_level: passive
• log_level_replica: warning
• log_on_each_node: True
• logging_nan_inf_filter: True
• save_safetensors: True
• save_on_each_node: False
• save_only_model: False
• restore_callback_states_from_checkpoint: False
• no_cuda: False
• use_cpu: False
• use_mps_device: False
• seed: 42
• data_seed: None
• jit_mode_eval: False
• use_ipex: False
• bf16: False
• fp16: False
• fp16_opt_level: O1
• half_precision_backend: auto
• bf16_full_eval: False
• fp16_full_eval: False
• tf32: None
• local_rank: 0
• ddp_backend: None
• tpu_num_cores: None
• tpu_metrics_debug: False
• debug: []
• dataloader_drop_last: False
• dataloader_num_workers: 0
• dataloader_prefetch_factor: None
• past_index: -1
• disable_tqdm: False
• remove_unused_columns: True
• label_names: None
• load_best_model_at_end: False
• ignore_data_skip: False
• fsdp: []
• fsdp_min_num_params: 0
• fsdp_config: {'min_num_params': 0, 'xla': False, 'xla_fsdp_v2': False, 'xla_fsdp_grad_ckpt': False}
• fsdp_transformer_layer_cls_to_wrap: None
• accelerator_config: {'split_batches': False, 'dispatch_batches': None, 'even_batches': True, 'use_seedable_sampler': True, 'non_blocking': False, 'gradient_accumulation_kwargs': None}
• deepspeed: None
• label_smoothing_factor: 0.0
• optim: adamw_torch
• optim_args: None
• adafactor: False
• group_by_length: False
• length_column_name: length
• ddp_find_unused_parameters: None
• ddp_bucket_cap_mb: None
• ddp_broadcast_buffers: False
• dataloader_pin_memory: True
• dataloader_persistent_workers: False
• skip_memory_metrics: True
• use_legacy_prediction_loop: False
• push_to_hub: False
• resume_from_checkpoint: None
• hub_model_id: None
• hub_strategy: every_save
• hub_private_repo: None
• hub_always_push: False
• gradient_checkpointing: False
• gradient_checkpointing_kwargs: None
• include_inputs_for_metrics: False
• include_for_metrics: []
• eval_do_concat_batches: True
• fp16_backend: auto
• push_to_hub_model_id: None
• push_to_hub_organization: None
• mp_parameters:
• auto_find_batch_size: False
• full_determinism: False
• torchdynamo: None
• ray_scope: last
• ddp_timeout: 1800
• torch_compile: False
• torch_compile_backend: None
• torch_compile_mode: None
• dispatch_batches: None
• split_batches: None
• include_tokens_per_second: False
• include_num_input_tokens_seen: False
• neftune_noise_alpha: None
• optim_target_modules: None
• batch_eval_metrics: False
• eval_on_start: False
• use_liger_kernel: False
• eval_use_gather_object: False
• average_tokens_across_devices: False
• prompts: None
• batch_sampler: batch_sampler
• multi_dataset_batch_sampler: round_robin

</details>

Training Logs

Framework Versions

• Python: 3.12.9
• Sentence Transformers: 3.4.1
• Transformers: 4.48.3
• PyTorch: 2.5.1+cu124
• Accelerate: 1.3.0
• Datasets: 3.2.0
• Tokenizers: 0.21.0

Citation

BibTeX

Sentence Transformers

@inproceedings{reimers-2019-sentence-bert,
    title = "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks",
    author = "Reimers, Nils and Gurevych, Iryna",
    booktitle = "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing",
    month = "11",
    year = "2019",
    publisher = "Association for Computational Linguistics",
    url = "https://arxiv.org/abs/1908.10084",
}

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