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How does the performance of our method state-of-the-art?

How does the performance of our method compared to the existing state-of-the-art?

May I know which one is grammatical?

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    Neither of these sentences is grammatical--they have no lexical verb, only the auxiliary do, and state-of-the-art can act only as an attributive adjective. I suspect what you want is How does our method compare to existing state-of-the-art methods? Mar 15 '14 at 14:13
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Neither I'm afraid. You're lacking some grammar aside from your question.

Try something like this:

How does the performance of our method compare against existing state-of-the-art methods?

When you're using a verb in an object question (like yours is), you need to use an auxiliary verb, in this case "does". If you use it in present, "does + subject + compare", or if you wish to use it in the past, "did". "How did the performance of our method compare..?"

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