Dynamical properties of profinite actions

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($\tau$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to answering the question of Lubotzky and Zuk, whether for families of subgroups, property ($\tau$) is inherited to the lattice of subgroups generated by the family. On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicite estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expander covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.Comment: Corrections made based on the referee's comment

On the intersection conjecture for infinite trees of matroids

Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2-separations into finite parts

Robustness: a New Form of Heredity Motivated by Dynamic Networks

We investigate a special case of hereditary property in graphs, referred to as {\em robustness}. A property (or structure) is called robust in a graph $G$ if it is inherited by all the connected spanning subgraphs of $G$. We motivate this definition using two different settings of dynamic networks. The first corresponds to networks of low dynamicity, where some links may be permanently removed so long as the network remains connected. The second corresponds to highly-dynamic networks, where communication links appear and disappear arbitrarily often, subject only to the requirement that the entities are temporally connected in a recurrent fashion ({\it i.e.} they can always reach each other through temporal paths). Each context induces a different interpretation of the notion of robustness. We start by motivating the definition and discussing the two interpretations, after what we consider the notion independently from its interpretation, taking as our focus the robustness of {\em maximal independent sets} (MIS). A graph may or may not admit a robust MIS. We characterize the set of graphs \forallMIS in which {\em all} MISs are robust. Then, we turn our attention to the graphs that {\em admit} a robust MIS (\existsMIS). This class has a more complex structure; we give a partial characterization in terms of elementary graph properties, then a complete characterization by means of a (polynomial time) decision algorithm that accepts if and only if a robust MIS exists. This algorithm can be adapted to construct such a solution if one exists

Certified Computation from Unreliable Datasets

A wide range of learning tasks require human input in labeling massive data. The collected data though are usually low quality and contain inaccuracies and errors. As a result, modern science and business face the problem of learning from unreliable data sets. In this work, we provide a generic approach that is based on \textit{verification} of only few records of the data set to guarantee high quality learning outcomes for various optimization objectives. Our method, identifies small sets of critical records and verifies their validity. We show that many problems only need $\text{poly}(1/\varepsilon)$ verifications, to ensure that the output of the computation is at most a factor of $(1 \pm \varepsilon)$ away from the truth. For any given instance, we provide an \textit{instance optimal} solution that verifies the minimum possible number of records to approximately certify correctness. Then using this instance optimal formulation of the problem we prove our main result: "every function that satisfies some Lipschitz continuity condition can be certified with a small number of verifications". We show that the required Lipschitz continuity condition is satisfied even by some NP-complete problems, which illustrates the generality and importance of this theorem. In case this certification step fails, an invalid record will be identified. Removing these records and repeating until success, guarantees that the result will be accurate and will depend only on the verified records. Surprisingly, as we show, for several computation tasks more efficient methods are possible. These methods always guarantee that the produced result is not affected by the invalid records, since any invalid record that affects the output will be detected and verified