All reports by Author Michael Forbes:

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TR16-045
| 22nd March 2016
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Michael Forbes, Mrinal Kumar, Ramprasad Saptharishi#### Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

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TR15-184
| 21st November 2015
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Matthew Anderson, Michael Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk#### Identity Testing and Lower Bounds for Read-$k$ Oblivious Algebraic Branching Programs

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TR14-162
| 28th November 2014
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Michael Forbes, Venkatesan Guruswami#### Dimension Expanders via Rank Condensers

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TR13-132
| 23rd September 2013
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Michael Forbes, Ramprasad Saptharishi, Amir Shpilka#### Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order

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TR13-033
| 1st March 2013
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Michael Forbes, Amir Shpilka#### Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing

Revisions: 1

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TR12-115
| 11th September 2012
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Michael Forbes, Amir Shpilka#### Quasipolynomial-time Identity Testing of Non-Commutative and Read-Once Oblivious Algebraic Branching Programs

Revisions: 1

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TR11-147
| 2nd November 2011
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Michael Forbes, Amir Shpilka#### On Identity Testing of Tensors, Low-rank Recovery and Compressed Sensing

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TR11-110
| 10th August 2011
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Alessandro Chiesa, Michael Forbes#### Improved Soundness for QMA with Multiple Provers

Revisions: 1

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TR11-010
| 1st February 2011
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Boris Alexeev, Michael Forbes, Jacob Tsimerman#### Tensor Rank: Some Lower and Upper Bounds

Michael Forbes, Mrinal Kumar, Ramprasad Saptharishi

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P \in F[x_1, x_2, \ldots, x_n]$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to {syntactically} computing $P$, when $C \equiv P$ as formal polynomials. In this ... more >>>

Matthew Anderson, Michael Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk

Read-$k$ oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs).

In this work, we give an exponential lower bound of $\exp(n/k^{O(k)})$ on the width of any read-$k$ oblivious ABP computing some explicit multilinear polynomial $f$ that is computed by a ...
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Michael Forbes, Venkatesan Guruswami

An emerging theory of "linear-algebraic pseudorandomness" aims to understand the linear-algebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension ... more >>>

Michael Forbes, Ramprasad Saptharishi, Amir Shpilka

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n^(lg^2 n) time. Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the ... more >>>

Michael Forbes, Amir Shpilka

Mulmuley recently gave an explicit version of Noether's Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In ... more >>>

Michael Forbes, Amir Shpilka

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing (PIT) algorithms for read-once oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but prior to this work had no known such black-box algorithm. Here we ... more >>>

Michael Forbes, Amir Shpilka

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but has no known such black-box algorithm. We recast this problem as ... more >>>

Alessandro Chiesa, Michael Forbes

We present three contributions to the understanding of QMA with multiple provers:

1) We give a tight soundness analysis of the protocol of [Blier and Tapp, ICQNM '09], yielding a soundness gap $\Omega(N^{-2})$, which is the best-known soundness gap for two-prover QMA protocols with logarithmic proof size. Maybe ...
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Boris Alexeev, Michael Forbes, Jacob Tsimerman

The results of Strassen and Raz show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds.

We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T:[n]^d->F with rank at least 2n^(floor(d/2))+n-Theta(d log n). ... more >>>