This is a sentence-transformers model finetuned from answerdotai/ModernBERT-base. It maps sentences & paragraphs to a 768-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: answerdotai/ModernBERT-base
Maximum Sequence Length: 8192 tokens
Output Dimensionality: 768 dimensions
Similarity Function: Cosine Similarity

Language: en
License: apache-2.0

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': 8192, 'do_lower_case': False}) with Transformer model: ModernBertModel 
  (1): Pooling({'word_embedding_dimension': 768, '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.

python

from sentence_transformers import SentenceTransformer

# Download from the 🤗 Hub
model = SentenceTransformer("Master-thesis-NAP/ModernBERT-base-finetuned-hard_negatives")
# Run inference
sentences = [
    '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?',
    '(\\cite{DangWangXieZhou})\\label{Theorem_Error_Estimate_k}\nLet 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:\n\\begin{eqnarray*}\na(F_{k,h}^{(\\ell)}w, u_{i,h}^{(\\ell)}) = a(w, u_{i,h}^{(\\ell)}), \\ \\ \\ i=1, \\cdots, k\\ \\ {\\rm for}\\ w\\in V.\n\\end{eqnarray*}\nThen 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:\n\\begin{eqnarray*}\\label{Error_Estimate_Inverse}\n \\left\\|\\bar u_{i,h} - F_{k,h}^{(\\ell+1)}\\bar u_{i,h} \\right\\|_a \\leq\n \\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}}\n\\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,\n\\end{eqnarray*}\nwhere $\\delta_{k,i,h}^{(\\ell)} $ is defined as follows:\n\\begin{eqnarray*}\n\\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.\n\\end{eqnarray*}\nFurthermore, the following $\\left\\|\\cdot\\right\\|_b$-norm error estimate holds:\n\\begin{eqnarray*}\n\\left\\|\\bar u_{i,h} -F_{k,h}^{(\\ell+1)}\\bar u_{i,h} \\right\\|_b\\leq \n\\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.\n\\end{eqnarray*}',
    '\\big[{\\bf Condition $SD1(h)$}\\big]\\label{DefnSD1(h)}\n\nIn \\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. \n\nFor example, let us view Step C of the approximation algorithm described in Section 5.2 of \\cite{MDL}. \nIt 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.\nThis 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.',
]
embeddings = model.encode(sentences)
print(embeddings.shape)
# [3, 768]

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

Evaluation

Metrics

Information Retrieval

Dataset: TESTING
Evaluated with InformationRetrievalEvaluator

Metric	Value
cosine_accuracy@1	0.8599
cosine_accuracy@3	0.9113
cosine_accuracy@5	0.9272
cosine_accuracy@10	0.9432
cosine_precision@1	0.8599
cosine_precision@3	0.6012
cosine_precision@5	0.4809
cosine_precision@10	0.3352
cosine_recall@1	0.0415
cosine_recall@3	0.0813
cosine_recall@5	0.1041
cosine_recall@10	0.1366
cosine_ndcg@10	0.4373
cosine_mrr@10	0.8887
cosine_map@100	0.1552

Training Details

Training Dataset

Unnamed Dataset

Size: 79,876 training samples
Columns: anchor, positive, and negative
Approximate statistics based on the first 1000 samples:
anchor positive negative
type string string string
details
min: 9 tokens
mean: 38.48 tokens
max: 142 tokens
min: 5 tokens
mean: 210.43 tokens
max: 924 tokens
min: 14 tokens
mean: 88.74 tokens
max: 516 tokens

Samples:

anchor	positive	negative
`What is the limit of the proportion of 1's in the sequence $a_n$ as $n$ approaches infinity, given that $0 \leq 3g_n -2n \leq 4$?`	`Let $g_n$ be the number of $1$'s in the sequence $a_1 a_2 \cdots a_n$.[object Object]Then [object Object]\begin{equation}[object Object]0 \leq 3g_n -2n \leq 4[object Object]\label{star}[object Object]\end{equation}[object Object]for all $n$, and hence[object Object]$\lim_{n \rightarrow \infty} g_n/n = 2/3$.[object Object]\label{thm1}`	`[] The tree $ \mathcal{T}_{ \infty}$ is the limit in Benjamini--Schramm quenched sense of $ \mathbb{T}_n$ as $n$ goes to infinity.`
`Does the statement of \textbf{ThmConjAreTrue} imply that the maximum genus of a locally Cohen-Macaulay curve in $\mathbb{P}^3_{\mathbb{C}}$ of degree $d$ that does not lie on a surface of degree $s-1$ is always equal to $g(d,s)$?`	`\label{ThmConjAreTrue}[object Object]Conjectures \ref{Conj1} and \ref{Conj2} are true.[object Object]As a consequence, [object Object]if either $d=s \geq 1$ or $d \geq 2s+1 \geq 3$, [object Object]the maximum genus of a locally Cohen-Macaulay curve in $\mathbb{P}^3_{\mathbb{C}}$ of degree $d$ that does not lie on a surface of degree $s-1$ is equal to $g(d,s)$.`	`[{\cite[Corollary 2.2.2 with $p=3$]{BSY}}][object Object] Let $S$ be a non-trivial Severi-Brauer surface over a perfect field $\textbf{k}$. Then $S$ does not contain points of degree $d$, where $d$ is not divisible by $3$. On the other hand $S$ contains a point of degree $3$.`
`\emph{Is the statement \emph{If $X$ is a compact Hausdorff space, then $X$ is normal}, proven in the first isomorphism theorem for topological groups, or is it a well-known result in topology?}`	`}[object Object]\newcommand{\ep}{`	`\label{prop:coherence}[object Object] If $X$ is a qcqs scheme, then $RX$ is coherent in the sense that the set of quasi-compact open subsets of $RX$ is closed under finite intersections and forms a basis for the topology of $RX$.`

Loss: MultipleNegativesRankingLoss with these parameters:
json
1{ 2 "scale": 20.0, 3 "similarity_fct": "cos_sim" 4}

Training Hyperparameters

Non-Default Hyperparameters

eval_strategy: epoch
per_device_train_batch_size: 16
per_device_eval_batch_size: 16
gradient_accumulation_steps: 8
learning_rate: 2e-05
num_train_epochs: 4
lr_scheduler_type: cosine
warmup_ratio: 0.1
bf16: True
tf32: True
load_best_model_at_end: True
optim: adamw_torch_fused
batch_sampler: no_duplicates

All Hyperparameters

Click to expand

overwrite_output_dir: False
do_predict: False
eval_strategy: epoch
prediction_loss_only: True
per_device_train_batch_size: 16
per_device_eval_batch_size: 16
per_gpu_train_batch_size: None
per_gpu_eval_batch_size: None
gradient_accumulation_steps: 8
eval_accumulation_steps: None
torch_empty_cache_steps: None
learning_rate: 2e-05
weight_decay: 0.0
adam_beta1: 0.9
adam_beta2: 0.999
adam_epsilon: 1e-08
max_grad_norm: 1.0
num_train_epochs: 4
max_steps: -1
lr_scheduler_type: cosine
lr_scheduler_kwargs: {}
warmup_ratio: 0.1
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: True
fp16: False
fp16_opt_level: O1
half_precision_backend: auto
bf16_full_eval: False
fp16_full_eval: False
tf32: True
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: True
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_fused
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
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: no_duplicates
multi_dataset_batch_sampler: proportional

Training Logs

Click to expand

Epoch	Step	Training Loss	TESTING_cosine_ndcg@10
0.0160	10	28.2978	-
0.0320	20	27.3071	-
0.0481	30	26.0992	-
0.0641	40	24.4038	-
0.0801	50	21.5478	-
0.0961	60	16.4121	-
0.1122	70	11.5764	-
0.1282	80	7.7549	-
0.1442	90	5.8755	-
0.1602	100	3.9385	-
0.1762	110	2.7828	-
0.1923	120	2.1444	-
0.2083	130	1.7805	-
0.2243	140	1.5387	-
0.2403	150	1.4621	-
0.2564	160	1.3322	-
0.2724	170	1.333	-
0.2884	180	1.1738	-
0.3044	190	0.9632	-
0.3204	200	0.9102	-
0.3365	210	1.0278	-
0.3525	220	0.883	-
0.3685	230	0.9874	-
0.3845	240	0.8836	-
0.4006	250	0.8743	-
0.4166	260	0.6613	-
0.4326	270	0.774	-
0.4486	280	0.6984	-
0.4647	290	0.6666	-
0.4807	300	0.6525	-
0.4967	310	0.6591	-
0.5127	320	0.6595	-
0.5287	330	0.5803	-
0.5448	340	0.6166	-
0.5608	350	0.6912	-
0.5768	360	0.6861	-
0.5928	370	0.7161	-
0.6089	380	0.6541	-
0.6249	390	0.6161	-
0.6409	400	0.4908	-
0.6569	410	0.4949	-
0.6729	420	0.5433	-
0.6890	430	0.5089	-
0.7050	440	0.5373	-
0.7210	450	0.5043	-
0.7370	460	0.4508	-
0.7531	470	0.6151	-
0.7691	480	0.4508	-
0.7851	490	0.4458	-
0.8011	500	0.3999	-
0.8171	510	0.4081	-
0.8332	520	0.3537	-
0.8492	530	0.5236	-
0.8652	540	0.5618	-
0.8812	550	0.4466	-
0.8973	560	0.4007	-
0.9133	570	0.4155	-
0.9293	580	0.3989	-
0.9453	590	0.7077	-
0.9613	600	0.423	-
0.9774	610	0.5218	-
0.9934	620	0.4137	-
1.0	625	-	0.4165
1.0080	630	0.4212	-
1.0240	640	0.3114	-
1.0401	650	0.2661	-
1.0561	660	0.215	-
1.0721	670	0.1974	-
1.0881	680	0.224	-
1.1041	690	0.2187	-
1.1202	700	0.2888	-
1.1362	710	0.2571	-
1.1522	720	0.2582	-
1.1682	730	0.2777	-
1.1843	740	0.2297	-
1.2003	750	0.2173	-
1.2163	760	0.2797	-
1.2323	770	0.2019	-
1.2483	780	0.2301	-
1.2644	790	0.2282	-
1.2804	800	0.2131	-
1.2964	810	0.2593	-
1.3124	820	0.287	-
1.3285	830	0.1675	-
1.3445	840	0.1342	-
1.3605	850	0.2494	-
1.3765	860	0.2151	-
1.3925	870	0.2111	-
1.4086	880	0.2471	-
1.4246	890	0.1846	-
1.4406	900	0.1864	-
1.4566	910	0.1998	-
1.4727	920	0.2894	-
1.4887	930	0.2177	-
1.5047	940	0.2325	-
1.5207	950	0.2442	-
1.5368	960	0.2016	-
1.5528	970	0.2012	-
1.5688	980	0.2357	-
1.5848	990	0.2224	-
1.6008	1000	0.3016	-
1.6169	1010	0.2209	-
1.6329	1020	0.1773	-
1.6489	1030	0.2279	-
1.6649	1040	0.2297	-
1.6810	1050	0.1896	-
1.6970	1060	0.2662	-
1.7130	1070	0.254	-
1.7290	1080	0.2416	-
1.7450	1090	0.1875	-
1.7611	1100	0.2494	-
1.7771	1110	0.1665	-
1.7931	1120	0.1605	-
1.8091	1130	0.2083	-
1.8252	1140	0.2155	-
1.8412	1150	0.1671	-
1.8572	1160	0.1846	-
1.8732	1170	0.1882	-
1.8892	1180	0.214	-
1.9053	1190	0.2179	-
1.9213	1200	0.242	-
1.9373	1210	0.2511	-
1.9533	1220	0.1902	-
1.9694	1230	0.2275	-
1.9854	1240	0.1938	-
2.0	1250	0.1702	0.4336
2.0160	1260	0.0784	-
2.0320	1270	0.0783	-
2.0481	1280	0.0959	-
2.0641	1290	0.1046	-
2.0801	1300	0.092	-
2.0961	1310	0.0769	-
2.1122	1320	0.1121	-
2.1282	1330	0.1442	-
2.1442	1340	0.0715	-
2.1602	1350	0.0907	-
2.1762	1360	0.0769	-
2.1923	1370	0.0753	-
2.2083	1380	0.0792	-
2.2243	1390	0.0716	-
2.2403	1400	0.0726	-
2.2564	1410	0.0909	-
2.2724	1420	0.0843	-
2.2884	1430	0.0857	-
2.3044	1440	0.0736	-
2.3204	1450	0.0624	-
2.3365	1460	0.1016	-
2.3525	1470	0.0729	-
2.3685	1480	0.0859	-
2.3845	1490	0.0677	-
2.4006	1500	0.0696	-
2.4166	1510	0.0796	-
2.4326	1520	0.0698	-
2.4486	1530	0.0773	-
2.4647	1540	0.0977	-
2.4807	1550	0.0869	-
2.4967	1560	0.0792	-
2.5127	1570	0.1235	-
2.5287	1580	0.0733	-
2.5448	1590	0.061	-
2.5608	1600	0.0859	-
2.5768	1610	0.082	-
2.5928	1620	0.0829	-
2.6089	1630	0.0833	-
2.6249	1640	0.0831	-
2.6409	1650	0.1032	-
2.6569	1660	0.079	-
2.6729	1670	0.0645	-
2.6890	1680	0.0798	-
2.7050	1690	0.0963	-
2.7210	1700	0.0717	-
2.7370	1710	0.1132	-
2.7531	1720	0.0715	-
2.7691	1730	0.0825	-
2.7851	1740	0.0826	-
2.8011	1750	0.0762	-
2.8171	1760	0.0614	-
2.8332	1770	0.0944	-
2.8492	1780	0.0811	-
2.8652	1790	0.0852	-
2.8812	1800	0.0989	-
2.8973	1810	0.078	-
2.9133	1820	0.0696	-
2.9293	1830	0.0897	-
2.9453	1840	0.1004	-
2.9613	1850	0.1178	-
2.9774	1860	0.0787	-
2.9934	1870	0.056	-
3.0	1875	-	0.4324
3.0080	1880	0.0632	-
3.0240	1890	0.0398	-
3.0401	1900	0.0352	-
3.0561	1910	0.0634	-
3.0721	1920	0.0472	-
3.0881	1930	0.0744	-
3.1041	1940	0.0595	-
3.1202	1950	0.0551	-
3.1362	1960	0.0451	-
3.1522	1970	0.0671	-
3.1682	1980	0.0364	-
3.1843	1990	0.0459	-
3.2003	2000	0.0533	-
3.2163	2010	0.0512	-
3.2323	2020	0.066	-
3.2483	2030	0.0667	-
3.2644	2040	0.0597	-
3.2804	2050	0.0556	-
3.2964	2060	0.0453	-
3.3124	2070	0.0677	-
3.3285	2080	0.0622	-
3.3445	2090	0.0587	-
3.3605	2100	0.0466	-
3.3765	2110	0.053	-
3.3925	2120	0.0505	-
3.4086	2130	0.0523	-
3.4246	2140	0.061	-
3.4406	2150	0.043	-
3.4566	2160	0.0513	-
3.4727	2170	0.0574	-
3.4887	2180	0.0506	-
3.5047	2190	0.0674	-
3.5207	2200	0.0572	-
3.5368	2210	0.0565	-
3.5528	2220	0.0671	-
3.5688	2230	0.0682	-
3.5848	2240	0.0537	-
3.6008	2250	0.0575	-
3.6169	2260	0.0584	-
3.6329	2270	0.0584	-
3.6489	2280	0.041	-
3.6649	2290	0.0529	-
3.6810	2300	0.0657	-
3.6970	2310	0.0416	-
3.7130	2320	0.0516	-
3.7290	2330	0.0651	-
3.7450	2340	0.0767	-
3.7611	2350	0.0432	-
3.7771	2360	0.0496	-
3.7931	2370	0.0543	-
3.8091	2380	0.053	-
3.8252	2390	0.0599	-
3.8412	2400	0.0528	-
3.8572	2410	0.0449	-
3.8732	2420	0.0461	-
3.8892	2430	0.0351	-
3.9053	2440	0.0385	-
3.9213	2450	0.048	-
3.9373	2460	0.0514	-
3.9533	2470	0.0495	-
3.9694	2480	0.0597	-
3.9854	2490	0.0523	-
4.0	2500	0.049	0.4373

The bold row denotes the saved checkpoint.

Framework Versions

Python: 3.11.12
Sentence Transformers: 4.1.0
Transformers: 4.52.2
PyTorch: 2.6.0+cu124
Accelerate: 1.7.0
Datasets: 2.14.4
Tokenizers: 0.21.1

Citation

BibTeX

Sentence Transformers

bibtex

@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",
}

MultipleNegativesRankingLoss

bibtex

@misc{henderson2017efficient,
    title={Efficient Natural Language Response Suggestion for Smart Reply},
    author={Matthew Henderson and Rami Al-Rfou and Brian Strope and Yun-hsuan Sung and Laszlo Lukacs and Ruiqi Guo and Sanjiv Kumar and Balint Miklos and Ray Kurzweil},
    year={2017},
    eprint={1705.00652},
    archivePrefix={arXiv},
    primaryClass={cs.CL}
}

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